2.1 Decorated trees and frequency interactions

We consider a set of decorated trees following the formalism developed in [Reference Bruned, Hairer and Zambotti13]. These trees will encode the Fourier coefficients of the numerical scheme.

We assume a finite set $ {\mathfrak {L}}$ and frequencies $ k_1,\ldots ,k_n \in {\mathbf {Z}}^{d} $ . The set $ {\mathfrak {L}}$ parametrises a set of differential operators with constant coefficients, whose symbols are given by the polynomials $ (P_{{\mathfrak {t}}})_{{\mathfrak {t}} \in {\mathfrak {L}}} $ . We define the set of decorated trees $ \hat {\mathcal {T}} $ as elements of the form $ T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}} = (T,{\mathfrak {n}},{\mathfrak {f}},{\mathfrak {e}}) $ where

• • $ T $ is a nonplanar rooted tree with root $ \varrho _T $ , node set $N_T$ and edge set $E_T$ . We denote the leaves of $ T $ by $ L_T $ . $ T $ must also be a planted tree, which means that there is only one edge outgoing the root.

• • The map $ {\mathfrak {e}} : E_T \rightarrow {\mathfrak {L}} \times \lbrace 0,1\rbrace $ are edge decorations.

• • The map $ {\mathfrak {n}} : N_T \setminus \lbrace \varrho _T \rbrace \rightarrow \mathbf {N} $ are node decorations. For every inner node $ v$ , this map encodes a monomial of the form $ \xi ^{{\mathfrak {n}}(v)} $ where $ \xi $ is a time variable.

• • The map $ {\mathfrak {f}} : N_T \setminus \lbrace \varrho _T \rbrace \rightarrow {\mathbf {Z}}^{d}$ are node decorations. These decorations are frequencies that satisfy for every inner node $ u $ :

  (43) $$ \begin{align} (-1)^{\mathfrak{p}(e_u)}{\mathfrak{f}}(u) = \sum_{e=(u,v) \in E_T} (-1)^{\mathfrak{p}(e)} {\mathfrak{f}}(v) \end{align} $$
  where $ {\mathfrak {e}}(e) = ({\mathfrak {t}}(e),\mathfrak {p}(e)) $ and $ e_u $ is the edge outgoing $ u $ of the form $ (v,u) $ . From this definition, one can see that the node decorations $ ({\mathfrak {f}}(u))_{u \in L_T} $ determine the decoration of the inner nodes. We assume that the node decorations at the leaves are linear combinations of the $ k_i $ with coefficients in $ \lbrace -1,0,1 \rbrace $ .

• • We assume that the root of $ T $ has no decoration.

When the node decoration $ {\mathfrak {n}} $ is zero, we will denote the decorated trees $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} $ as $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} = (T,{\mathfrak {f}},{\mathfrak {e}}) $ . The set of decorated trees satisfying such a condition is denoted by $ \hat {\mathcal {T}}_0 $ . We say that $ \bar T_{\bar {\mathfrak {e}}}^{\bar {\mathfrak {f}}} $ is a decorated subtree of $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} \in \hat {\mathcal {T}}_0 $ if $ \bar T $ is a subtree of $ T $ and the restriction of the decorations $ {\mathfrak {f}}, {\mathfrak {e}} $ of $ T $ to $ \bar T $ are given by $ \bar {\mathfrak {f}} $ and $ \bar {\mathfrak {e}} $ . Notice that because trees considered in this framework are always planted, we look only at subtrees that are planted. A planted subtree of $ T $ is of the form $ T_e $ where $ e \in E_T $ and $ T_e $ corresponds to the tree above $ e $ . The nodes of $ T_e $ are given by all of the nodes whose path to the root contains $ e $ .

Example 5. Below, we give an example of a decorated tree $ T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}} $ where the edges are labelled with numbers from $ 1 $ to $ 7 $ and the set $ N_T \setminus \lbrace \varrho _T\rbrace $ is labelled by $\lbrace a,b,c,d,e,f,g \rbrace $ :

We denote by $\hat H $ (respectively $ \hat H_0 $ ) the (unordered) forests composed of trees in $ \hat {\mathcal {T}} $ (respectively $ \hat {\mathcal {T}}_0 $ ; including the empty forest denoted by ${\mathbf {1}}$ ). Their linear spans are denoted by $\hat {\mathcal {H}} $ and $ \hat {\mathcal {H}}_0 $ . We extend the definition of decorated subtrees to forests by saying that $ T $ is a decorated subtree of the decorated forest $ F $ if their exists a decorated tree $ \bar T$ in $ F $ such that $ T $ is a decorated subtree of $ \bar T $ . The forest product is denoted by $ \cdot $ and the counit is $ {\mathbf {1}}^{\star } $ which is nonzero only on the empty forest.

In order to represent these decorated trees, we introduce a symbolic notation. An edge decorated by $ o = ({\mathfrak {t}},{\mathfrak {p}}) $ is denoted by $ {\cal I}_{o} $ . The symbol $ {\cal I}_{o}(\lambda _{k}^{\ell } \cdot ) : \hat {\mathcal {H}} \rightarrow \hat {\mathcal {H}} $ is viewed as the operation that merges all of the roots of the trees composing the forest into one node decorated by $(\ell ,k) \in \mathbf {N} \times {\mathbf {Z}}^{d} $ . We obtain a decorated tree which is then grafted onto a new root with no decoration. If the condition (43) is not satisfied on the argument, then ${\cal I}_{o}( \lambda _{k}^{\ell } \cdot )$ gives zero. If $ \ell = 0 $ , then the term $ \lambda _{k}^{\ell } $ is denoted by $ \lambda _{k} $ as a shorthand notation for $ \lambda _{k}^{0} $ . When $ \ell = 1 $ , it will be denoted by $ \lambda _{k}^{1} $ . The forest product between $ {\cal I}_{o_1}( \lambda ^{\ell _1}_{k_1}F_1) $ and $ {\cal I}_{o_2}( \lambda ^{\ell _2}_{k_2}F_2) $ is given by

$$ \begin{align*} {\cal I}_{o_1}( \lambda^{\ell_1}_{k_1} F_1) {\cal I}_{o_2}( \lambda^{\ell_2}_{k_2} F_2) := {\cal I}_{o_1}( \lambda^{\ell_1}_{k_1} F_1) \cdot {\cal I}_{o_2}( \lambda^{\ell_2}_{k_2} F_2). \end{align*} $$

Example 6. The following symbol

$$ \begin{align*} {\cal I}_{({\mathfrak{t}}(1),{\mathfrak{p}}(1))}( \lambda^{{\mathfrak{n}}(a)}_{{\mathfrak{f}}(a)}{\cal I}_{({\mathfrak{t}}(2),{\mathfrak{p}}(2))}( \lambda^{{\mathfrak{n}}(b)}_{{\mathfrak{f}}(b)}){\cal I}_{({\mathfrak{t}}(3),{\mathfrak{p}}(3))}( \lambda^{{\mathfrak{n}}(c)}_{{\mathfrak{f}}(c)})) \end{align*} $$

encodes the tree

(44)

We will see later (in Example 7) that the above tree with suitably chosen decorations describes the first iterated integral of the Kortweg–de Vries equation (3).

We are interested in the following quantity which represents the frequencies associated to this tree:

(45) $$ \begin{align} \mathscr{F}( T_{{\mathfrak{e}}}^{{\mathfrak{f}}} ) = \sum_{u \in N_T} P_{({\mathfrak{t}}(e_u),{\mathfrak{p}}(e_u))}({\mathfrak{f}}(u)) \end{align} $$

where $ e_u $ is the edge outgoing $ u $ of the form $ (v,u) $ and

(46) $$ \begin{align} P_{({\mathfrak{t}}(e_u),{\mathfrak{p}}(e_u))}({\mathfrak{f}}(u)){ \, := \,} (-1)^{{\mathfrak{p}}(e_u)}P_{{\mathfrak{t}}(e_u)}((-1)^{{\mathfrak{p}}(e_u)}{\mathfrak{f}}(u)). \end{align} $$

The term $ \mathscr {F}( T_{{\mathfrak {e}}}^{{\mathfrak {f}}}) $ has to be understood as a polynomial in multiple variables given by the $ k_i $ .

In the numerical scheme, what matters are the terms with maximal degree of frequency, which are here the monomials of higher degree; cf. $\mathcal {L}_{\tiny {\text {dom}}}$ . We compute them using the symbolic notation in the next section. We assume fixed $ {\mathfrak {L}}_{+} \subset {\mathfrak {L}} $ . This subset encodes integrals in time that are of the form $ \int _0^{\tau } e^{s P_{(\mathfrak {t},\mathfrak {p})}(\cdot )} \cdots ds $ for $ (\mathfrak {t},\mathfrak {p}) \in {\mathfrak {L}}_+ \times \lbrace 0,1 \rbrace $ ; see also its interpretation given in (82).

2.2 Dominant parts of trees and physical space maps

Definition 2.2. Let $ P(k_1,\ldots ,k_n) $ a polynomial in the $ k_i $ . If the higher monomials of $ P $ are of the form

$$ \begin{align*} a \sum_{i=1}^{n} (a_i k_i)^{m}, \quad a_i \in { \lbrace 0,1 \rbrace}, \, a \in {\mathbf{Z}}, \end{align*} $$

then we define $ {\mathcal {P}}_{\tiny {\text {dom}}}(P) $ as

(47) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}}(P) = a \left(\sum_{i=1}^{n} a_i k_i\right)^{m}. \end{align} $$

Otherwise, it is zero.

Remark 2.3. Given a polynomial $ P $ , one can compute its dominant part $ \mathcal {L}_{\text {dom}} $ and its lower part $\mathcal {L}_{\text {low}}$

$$ \begin{align*} \mathcal{L}_{\tiny{\text{dom}}} = {\mathcal{P}}_{\tiny{\text{dom}}}(P), \quad \mathcal{L}_{\tiny{\text{low}}} = \left({\mathrm{id}} - {\mathcal{P}}_{\tiny{\text{dom}}} \right)(P). \end{align*} $$

In our discretisation we will treat the dominant parts of the frequency interactions $\mathcal {L}_{\tiny {\text {dom}}}$ exactly, while approximating the lower-order parts $\mathcal {L}_{\tiny {\text {low}}} $ by Taylor series expansions (cf. also (12)). This will be achieved by applying recursively the operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ introduced in Definition 2.6.

Note that in the special case that ${\mathcal {P}}_{\tiny {\text {dom}}}(P) = 0$ , we have to expand all frequency interactions into a Taylor series. The latter, for instance, arises in the context of quadratic Schrödinger equations

(48) $$ \begin{align} i \partial_t u = -\Delta u + u^2, \quad u(0,x) = v(x) \end{align} $$

for which we face oscillations of type (cf. (10))

$$ \begin{align*} \int_0^\tau e^{i \xi \Delta } (e^{-i \xi \ \Delta} v)^2 d\xi & = \sum_{k_{1},k_{2}\in {\mathbf{Z}}^d}\hat{v}_{k_1}\hat{v}_{k_2} e^{i (k_1+k_2) x} \int_0^\tau e^{ -i \big(k_1+k_2\big)^2 \xi} e^{ i \big(k_1^2 + k_2^2 \big)\xi} d\xi \\ &=\sum_{k_{1},k_{2}\in {\mathbf{Z}}^d}\hat{v}_{k_1}\hat{v}_{k_2} e^{i (k_1+k_2) x} \int_0^\tau e^{ -2 i k_1 k_2 \xi} d\xi. \end{align*} $$

Here we recall the notation $ k \ell = k_1 \ell _1 + \ldots + k_d \ell _d $ for $k, \ell \in {\mathbf {Z}}^d$ . In contrast to the cubic NLS (21) where we have that (cf. (26))

$$ \begin{align*} { \mathcal{L}_{\text{dom}}(k_1) = 2k_1^2 \quad \text{and}\quad \mathcal{L}_{\text{low}}(k_1,k_2,k_3) = - 2 k_1 (k_2+k_3) + 2 k_2 k_3}, \end{align*} $$

we observe for the quadratic NLS (48) with the map $ {\mathcal {P}}_{\tiny {\text {dom}}} $ given in (47) that

$$ \begin{align*} P(k_1,k_2) & = - (k_1 + k_2)^2 + (k_1^2 + k_2^2) = - 2k_1 k_2, \quad {\mathcal{P}}_{\tiny{\text{dom}}}(P) = 0, \\ \mathcal{L}_{\tiny{\text{dom}}} & = {\mathcal{P}}_{\tiny{\text{dom}}}(P) =0, \quad \mathcal{L}_{\tiny{\text{low}}} = P - {\mathcal{P}}_{\tiny{\text{dom}}}(P) = - 2k_1 k_2. \end{align*} $$

Hence, although $\mathcal {L}= -\Delta $ and $\mathcal {L}_{\text {dom}}= 0 $ (which means that no oscillations are integrated exactly), we ‘only’ lose one derivative in the local approximation error (cf. (16)) as

$$ \begin{align*} \mathcal{O}\left(\tau^2 \mathcal{L}_{\text{low}}v\right)= \mathcal{O}\left(\tau^2 \vert \nabla \vert v\right). \end{align*} $$

Remark 2.4. Terms of type (47) will naturally arise when filtering out the dominant nonlinear frequency interactions in the PDE. We have to embed integrals over their exponentials into our discretisation. For their practical implementation it will therefore be essential to map fractions of (47) back to physical space.

Indeed, if we apply the inverse Fourier transform $ \mathcal {F}^{-1} $ , we get

(49) $$ \begin{align} \mathcal{F}^{-1} & \left( \sum_{\substack{0 \neq k=k_1 +\ldots +k_n\\ k_\ell \neq 0} } \frac{1}{(k_1 +\ldots +k_n)^m} \frac{1}{k_1^{m_1}} \ldots \frac{1}{k_n^{m_n}}v^1_{k_1}\ldots v^n_{k_n} e^{i kx} \right) \end{align} $$

$$\begin{align*}& = (-\Delta)^{- m/2} \prod_{\ell = 1}^n \left( (-\Delta)^{-m_\ell/2} v^\ell(x)\right) \end{align*}$$

where by abuse of notation we define the operator $(-\Delta )^{-1}$ in Fourier space as $(-\Delta )^{-1} f(x) = \sum _{ k \neq 0} \frac { \hat {f}_k }{k^2}e^{i k x}.$

In the next proposition, we elaborate on (49) and give a nice class of functions depending on the $k_i$ that we can map back to the physical space.

Proposition 2.5. Assume that we have polynomials $ Q $ in $ k_1, \ldots , k_n $ and $ k $ is a linear combination of the $k_i$ such that

$$ \begin{align*} Q & = \prod_j (\sum_{u \in V_j} a_{u,V_j} k_u )^{m_i}, \quad V_j \subset \lbrace 1,\ldots ,n \rbrace, \quad a_{u,V_j} \in \lbrace-1,1\rbrace, \\ k & = \sum_{u=1}^{n} a_u k_u, \quad a_u \in \lbrace -1,1 \rbrace, \end{align*} $$

where the $ V_i $ are either disjoint or if $ V_j \subset V_i $ we assume that there exist $ p_{i,j} $ such that

$$ \begin{align*} a_{u,V_i} = (-1)^{p_{i,j}} a_{u,V_j}, \quad u \in V_j. \end{align*} $$

We also suppose that the $ V_i $ are included in $ k $ in the sense that there exist $ p_{V_i} $ such that

$$ \begin{align*} a_u = (-1)^{p_{V_i}} a_{u,V_i}, \quad u \in V_i. \end{align*} $$

Then, one gets

$$ \begin{align*} \mathcal{F}^{-1} & \left( \sum_{ \substack{0 \neq k= a_1 k_1 +\ldots + a_n k_n\\ Q(k_1,\ldots ,k_n) \neq 0}} \frac{1}{Q} v^{1,a_1}_{k_1}\ldots v^{n,a_n}_{k_n} e^{i kx} \right)\\ & = \left(\prod_{V_i \subset V_j} (-1)^{p_{V_i}} (-\Delta)^{- m_i/2}_{V_i} \right) v^{1,a_1}\ldots v^{n,a_n} \end{align*} $$

where $ v^{i,1} = v^{i} $ and $ v^{i,-1} = \overline {v^{i}} $ . The operator $ (-\Delta )^{- m_i/2}_{V_i} $ acts only on the functions $\prod _{u \in V_i} v^{u,a_u} $ and the product starts by the smaller elements for the inclusion order.

Proof

We proceed by induction on the number of $ V_i $ . Let $ V_{\max } $ be an element among the $ V_i $ maximum for the inclusion order. Then, we get

$$ \begin{align*} \sum_{ \substack{0 \neq k= a_1 k_1 +\ldots + a_n k_n\\ Q(k_1,\ldots ,k_n) \neq 0}} & \frac{1}{Q} v^{1,a_1}_{k_1}\ldots v^{n,a_n}_{k_n} e^{i kx} = \sum_{\substack{0 \neq k= r + \ell \\ \ell \neq 0}} \frac{(-1)^{p_{V_{\max}}}}{\ell^{m_{\max}}}\sum_{ \substack{0 \neq r= \sum_{u \notin V_{\max}} a_u k_u \\ R \neq 0}} \frac{1}{R} \\ & \left( \prod_{j \notin V_{\max}} v^{j,a_j}_{k_j} \right) e^{i r x} \times \sum_{ \substack{0 \neq \ell= \sum_{u \in V_{\max}} a_u k_u \\ S \neq 0}} \frac{1}{S} \left( \prod_{j \in V_{\max}} v^{j,a_j}_{k_j} \right) e^{i \ell x} \end{align*} $$

where

$$ \begin{align*} S = \prod_{V_j \varsubsetneq V_{\max}} (\sum_{u \in V_j} a_{u,V_j} k_u )^{m_i}, \quad R = \prod_{V_j \cap V_{\max} = \emptyset} (\sum_{u \in V_j} a_{u,V_j} k_u )^{m_i}, \quad Q = R S \ell^{m_{max}}. \end{align*} $$

Thus, by applying the inverse Fourier transform, we get the term $ (-1)^{p_{V_{\max }}} (-\Delta )^{- m_{\max }/2}_{V_{\max }} $ from $ \frac {(-1)^{p_{V_{\max }}}}{\ell ^{m_{\max }}} $ . We conclude from the induction hypothesis on the two remaining sums.

The next definition will allow us to compute the dominant part of the frequency interactions of a given decorated forest in $ \hat H_0 $ . The idea is to filter out the dominant part using the operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ which selects the frequencies of highest order. The operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ only appears if we face an edge in $ {\mathfrak {L}}_+ $ which corresponds to an integral in time that we have to approximate.

Definition 2.6. We recursively define $\mathscr {F}_{\tiny {\text {dom}}}, \mathscr {F}_{\tiny {\text {low}}} : \hat H_{0} \rightarrow \mathbb {R}[{\mathbf {Z}}^d]$ as

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}({\mathbf{1}}) = 0 \quad \mathscr{F}_{\tiny{\text{dom}}}(F \cdot \bar F) & =\mathscr{F}_{\tiny{\text{dom}}}(F) + \mathscr{F}_{\tiny{\text{dom}}}(\bar F) \\ \mathscr{F}_{\tiny{\text{dom}}}\left( {\cal I}_{({\mathfrak{t}},{\mathfrak{p}})}( \lambda_{k}F) \right) & = \left\{ \begin{array}{l} \displaystyle {\mathcal{P}}_{\tiny{\text{dom}}}\left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(F) \right), \, \text{if } {\mathfrak{t}} \in {\mathfrak{L}}_+ , \\ P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(F), \quad \text{otherwise} \\ \end{array} \right. \\ \mathscr{F}_{\tiny{\text{low}}} \left( {\cal I}_{({\mathfrak{t}},{\mathfrak{p}})}( \lambda_{k}F) \right) & = \left( {\mathrm{id}} - {\mathcal{P}}_{\tiny{\text{dom}}} \right) \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(F) \right). \end{align*} $$

We extend these two maps to $ \hat H $ by ignoring the node decorations $ {\mathfrak {n}} $ .

Remark 2.7. The definition of $ {\mathcal {P}}_{\tiny {\text {dom}}} $ can be adapted depending on what is considered to be the dominant part. For example, if for $ {\mathfrak {t}}_2 \in {\mathfrak {L}}_+$ , one has (cf. (9))

$$ \begin{align*} P_{({\mathfrak{t}}_2,p)}( \lambda) = \frac{1}{\varepsilon^{\sigma}} + F_{({\mathfrak{t}}_2,p)}( \lambda). \end{align*} $$

Then we can define the dominant part only depending on $ \varepsilon $ (see Example 5.3):

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}}\left( P_{({\mathfrak{t}}_2,p)}(k) \right)= \frac{1}{\varepsilon^{\sigma}}. \end{align*} $$

The definition considers only the case where one does not have terms of the form $ 1/\varepsilon ^{\sigma } $ . It is sufficient for covering many interesting examples.

In the following we compute the dominant part $ \mathscr {F}_{\text {dom}}$ for the underlying trees of the cubic Schrödinger (2) and KdV (3) equation.

Example 7 (KdV). We consider the decorated tree $ T $ given in Example 44, where we fix the following decorations:

$$ \begin{align*} {\mathfrak{p}}(1) & = {\mathfrak{p}}(2) = {\mathfrak{p}}(3) = 0, \quad {\mathfrak{t}}(2) = {\mathfrak{t}}(3) = {\mathfrak{t}}_1, \quad {\mathfrak{t}}(1) = {\mathfrak{t}}_2, \\[-3pt] {\mathfrak{f}}(b) & = k_1, \quad {\mathfrak{f}}(c) = k_2, \quad {\mathfrak{f}}(a) = k_1 + k_2, \quad P_{{\mathfrak{t}}_2}( \lambda) = \lambda^3 , \quad P_{{\mathfrak{t}}_1}( \lambda) = - \lambda^3. \end{align*} $$

Now, we suppose $ {\mathfrak {L}}_+ = \lbrace {\mathfrak {t}}_2 \rbrace $ and $ {\mathfrak {L}} = \lbrace {\mathfrak {t}}_1, {\mathfrak {t}}_2 \rbrace $ . Then the tree (44) takes the form

(50)

This tree corresponds to the first iterated integral for the KdV equation (3). In more formal notation, we denote this tree by

(51)

where a blue edge encodes $ (\mathfrak {t}_2,0) $ and a brown edge is used for $(\mathfrak {t}_1,0)$ . The frequencies are given on the leaves. The ones on the inner nodes are determined by those on the leaves. On the left-hand side, we have given the symbolic notation. Together with Definition 2.6, one gets

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(T) & = {\mathcal{P}}_{\tiny{\text{dom}}} \left( (k_1+k_2)^{3} - k_1^{3} - k_2^{3} \right) = 0\\ \mathscr{F}_{\tiny{\text{low}}} (T) & = (k_1+k_2)^{3} - k_1^{3} - k_2^{3} = 3k_1 k_2 (k_1+k_2). \end{align*} $$

The fact that $ \mathscr {F}_{\tiny {\text {dom}}}(T) $ is zero comes from the fundamental choice of definition of the operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ in Definition 2.2.

Example 8 (Cubic Schrödinger). Next we consider the symbol

$$ \begin{align*} {\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{-k_1+k_2+k_3}{\cal I}_{({\mathfrak{t}}_2,1)}( \lambda_{k_1}){\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{k_2}){\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{k_3})) \end{align*} $$

with $P_{{\mathfrak {t}}_2}( \lambda )= \lambda ^2$ , $P_{{\mathfrak {t}}_1}( \lambda ) = - \lambda ^2$ , $ {\mathfrak {L}}_+ = \lbrace {\mathfrak {t}}_2 \rbrace $ and $ {\mathfrak {L}} = \lbrace {\mathfrak {t}}_1, {\mathfrak {t}}_2 \rbrace $ which encodes the tree

(52)

This tree corresponds to the frequency interaction of the first iterated integral for the cubic Schrödinger equation (2). In a more formal notation, we denote this tree by

(53)

where a blue edge encodes $ (\mathfrak {t}_2,0) $ , a brown edge is used for $ (\mathfrak {t}_1,0) $ and a dashed brown edge is for $ (\mathfrak {t}_2,1) $ . With Definition 2.6, we get

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(T) & = {\mathcal{P}}_{\tiny{\text{dom}}} \left( (-k_1+k_2+k_3)^{2} + (-k_1)^{2} - k_2^{2} - k_3^2 \right) \\& = {\mathcal{P}}_{\tiny{\text{dom}}} \left( 2k_1^2 - 2k_1(k_2+k_3) + 2 k_2 k_3\right) = 2 k_1^2\\ \mathscr{F}_{\tiny{\text{low}}} (T)& = (-k_1+k_2+k_3)^{2} + (-k_1)^{2} - k_2^{2} - k_3^2 - 2k_1^2\\ & = - 2k_1(k_2+k_3) + 2 k_2 k_3. \end{align*} $$

The map $ \mathscr {F}_{\tiny {\text {dom}}}$ has a nice property regarding the tree inclusions given in the next proposition. This inclusive property will be important in practical computations; see also Remark 1.3 and the examples in Section 5.

For Proposition 2.8, we need an additional assumption.

With this assumption, for a decorated forest $ F = \prod _i T_i $ such that $ {\mathcal {P}}_{\tiny {\text {dom}}}\left (\mathscr {F}_{\tiny {\text {dom}}}(F) \right ) \neq 0 $ , one has the nice identity

(54) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(F) \right) = {\mathcal{P}}_{\tiny{\text{dom}}}\left(\sum_i {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(T_i) \right) \right). \end{align} $$

We will illustrate this property and give a counterexample in an example below.

Example 9. We give a counterexample in the setting of the Schrödinger equation to (54) when ${\mathcal {P}}_{\tiny {\text {dom}}}\left (\mathscr {F}_{\tiny {\text {dom}}}(F) \right ) = 0 $ . We consider the following forest $ F = \prod _{i=1}^3 T_i$ where the decorated trees $ T_i $ are given by

Then, we can check Assumption 1 for $ F $ . One has

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(T_1) & = - k_4^2, \quad\mathscr{F}_{\tiny{\text{dom}}}(T_2) = - k_5^2, \quad\mathscr{F}_{\tiny{\text{dom}}}(T_3) = - (-k_1 + k_2 + k_3)^2 + 2 k_1^2, \\ \mathscr{F}_{\tiny{\text{dom}}}(F) & = - k_4^2 - k_5^2 - (-k_1 + k_2 + k_3)^2 + 2 k_1^2 \end{align*} $$

and

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}(T_1) \right) = - k_4^2, \quad {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_2) \right) = - k_5^2, \quad {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_3) \right) = 0. \end{align*} $$

Therefore, one obtains

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( \sum_{i=1}^3 {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_i) \right) \right) = - (k_4 + k_5)^2. \end{align*} $$

But, on the other hand,

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(F) \right) = 0. \end{align*} $$

Proposition 2.8. Let $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ be a decorated tree in $ \hat H_0 $ and $ e \in E_T $ . We recall that $ T_e $ corresponds to the subtree of $ T $ above $ e $ . The nodes of $ T_e $ are given by all of the nodes whose path to the root contains $ e $ . Under Assumption 1 one has

(55) $$ \begin{align} \begin{split} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_{{\mathfrak{e}}}^{{\mathfrak{f}}}) \right) & = a \left( \sum_{u \in V } a_u k_u \right)^{m}, \quad m \in \mathbf{N}, a \in {\mathbf{Z}}, \, V \subset L_T, \, a_u \in \lbrace -1,1 \rbrace \\ {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}((T_e)_{{\mathfrak{e}}}^{{\mathfrak{f}}}) \right) & = b \left( \sum_{u \in \bar V } b_u k_u \right)^{m_e}, \quad m_e \in \mathbf{N}, b \in {\mathbf{Z}}, \, \bar V \subset L_T, \, b_u \in \lbrace -1,1 \rbrace \end{split} \end{align} $$

and $\bar V \subset V \text { or } \bar V \cap V = \emptyset $ . If $ \bar V \subset V $ , then there exists $ \bar p \in \lbrace 0,1 \rbrace $ such that $ a_u = (-1)^{\bar p} b_u $ for every $ u \in \bar V $ .

Proof

We proceed by induction over the size of the decorated trees. We consider $T= {\cal I}_{({\mathfrak {t}},{\mathfrak {p}})}( \lambda _{k} F) $ where $ F = \prod _{i=1}^m T_i $ .

(i) If $F ={\mathbf {1}}$ , then one has

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) \right). \end{align*} $$

We conclude from the definition of $ {\mathcal {P}}_{\tiny {\text {dom}}} $ .

(ii) If $ m> 2 $ , then one gets

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + \sum_{i=1}^m \mathscr{F}_{\tiny{\text{dom}}}(T_i) \right). \end{align*} $$

Using Assumption 1, one obtains

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( \sum_{i=1}^m (P_{({\mathfrak{t}},{\mathfrak{p}})}(k^{(i)}) +\mathscr{F}_{\tiny{\text{dom}}}(T_i)) \right) \end{align*} $$

where $ k^{(i)} $ corresponds to the frequency attached to the node connected to the root of $ T_i $ . If $ {\mathcal {P}}_{\tiny {\text {dom}}} \left (\mathscr {F}_{\tiny {\text {dom}}}(T) \right ) \neq 0 $ , then from (54) we have

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( \sum_{i=1}^m {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k^{(i)}) +\mathscr{F}_{\tiny{\text{dom}}}(T_i) \right) \right). \end{align*} $$

We apply the induction hypothesis to each decorated tree $ \tilde T_i = {\cal I}_{({\mathfrak {t}},{\mathfrak {p}})}( \lambda _{k^{(i)}} T_i) $ and we recombine the various terms in order to conclude.

(iii) If $ m=1 $ , then $ F = {\cal I}_{({\mathfrak {t}}_1,{\mathfrak {p}}_1)}( \lambda _{ \bar k} T_1) $ where $ \bar k $ is equal to $ k $ up to a minus sign. We can assume without loss of generality that

(56) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(F) \right) = \mathscr{F}_{\tiny{\text{dom}}}(F). \end{align} $$

Indeed, otherwise,

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}(T)\right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + P_{({\mathfrak{t}}_1,{\mathfrak{p}}_1)}(\bar k) + \mathscr{F}_{\tiny{\text{dom}}}(T_1) \right). \end{align*} $$

We can see $ P_{({\mathfrak {t}},{\mathfrak {p}})}(k) + P_{({\mathfrak {t}}_1,{\mathfrak {p}}_1)}(\bar k) $ as a polynomial and then apply the induction hypothesis. We are down to the case (56) and we consider

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) & = {\mathcal{P}}_{\tiny{\text{dom}}}\left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) \right) \end{align*} $$

where now $ F = \bar T$ is just a decorated tree. We apply the induction hypothesis on $ \bar T $ and we get

(57) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) & = a \left( \sum_{u \in V } a_u k_u \right)^{m}, \quad a \in {\mathbf{Z}}, \, V \subset L_T, \, a_u \in \lbrace -1,1 \rbrace \\k & = \sum_{u \in L_T } c_u k_u, \quad c_{ u} = (-1)^{ p} a_{u} , \quad u \in V. \notag \end{align} $$

If the degree of $P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ is higher than the degree of $\mathscr {F}_{\tiny {\text {dom}}}(\bar T)$ , we obtain that

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) \right). \end{align*} $$

On the other hand, if the degree of $P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ is lower than the degree of $\mathscr {F}_{\tiny {\text {dom}}}(\bar T)$ ,

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right). \end{align*} $$

If $ P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ and $\mathscr {F}_{\tiny {\text {dom}}}(\bar T)$ have the same degree $ m $ , we get using the definition of $ P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ in (46) as well as the induction hypothesis on $\bar T$ given in (57) that

$$ \begin{align*} P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) & = \sum_{u \in V } \left( a (-1)^{p + m + {\mathfrak{p}}} + (- 1)^{{\mathfrak{p}}} \right)( (-1)^{{\mathfrak{p}}}c_u k_u)^{m} \\ & \quad{}+ \sum_{u \in L_T \setminus V } (- 1)^{{\mathfrak{p}}} ((-1)^{{\mathfrak{p}}} c_u k_u)^{m} + R \end{align*} $$

where $ R $ terms of lower orders.

By applying the map $ {\mathcal {P}}_{\tiny {\text {dom}}} $ defined in (47), we thus obtain an expression of the form

(58) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = b \left( \sum_{u \in \tilde{V} } (-1)^{{\mathfrak{p}}} c_u k_u \right)^{m} \,\text{for some } b \in {\mathbf{Z}} \end{align} $$

where $ \tilde V $ could be either $ L_T $ or $ L_T \setminus V $ .

Let $ e \in E_T $ , $ T_e \neq T $ , then $ T_e $ is a subtree of $ \bar T $ . By the induction hypothesis, one obtains (55), meaning that if we denote by $ V $ (respectively $ \bar V $ ) the set associated to $ \bar T $ (respectively $ T_e $ ), we get $ \bar V \subset V $ or $ \bar V \cap V = \emptyset $ .

In the first case, $\bar V \subset V $ , the assertion follows as $ V \subset \tilde {V} $ or $ V \cap \tilde {V} = \emptyset $ such that necessarily $\bar V \subset \tilde V$ or $\bar V \cap \tilde V = \emptyset $ .

In the second case, $ \bar V \cap V = \emptyset $ , we apply the induction hypothesis on $ T_e $ . Then, for $ v $ of $ T_e $ , there exists $ p $ such that the decoration $ {\mathfrak {f}}(v) $ is given by

$$ \begin{align*} {\mathfrak{f}}(v) = \sum_{u \in L_{T_v} } d_u k_u, \quad d_{ u} = (-1)^{ p} b_{u} , \quad u \in \bar V. \end{align*} $$

As $ {\mathfrak {f}}(v) $ appears as a subfactor in $ k $ , one has $ \bar V \subset L_T $ . Then, $ \bar {V} \cap V = \emptyset $ also gives that $ \bar V \subset L_T \setminus V $ . Therefore, we have $ \bar V \subset \tilde {V} $ , which concludes the proof.

Corollary 2.9. Let $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ a decorated tree in $ \hat {\mathcal {T}}_0 $ . We assume that Assumption 1 holds true for a set $ A $ of decorated subtrees of $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ such that ${\mathcal {P}}_{\tiny {\text {dom}}} \left (\mathscr {F}_{\tiny {\text {dom}}}(\bar T) \right ) =\mathscr {F}_{\tiny {\text {dom}}}(\bar T) \neq 0$ for $ \bar T \in A $ . Moreover, we assume that the $ \bar T $ are of the form $ (T_e)_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ where $ e \in E_T $ . Then, the following product

$$ \begin{align*} \prod_{\bar T\kern-1pt \in A} \frac{1}{\left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right)^{m_T}} \end{align*} $$

can be mapped back to physical space using operators of the form $(- \Delta )^{-m/2}_V $ as defined in Proposition 2.5.

Example 10. We illustrate Corollary 2.9 via an example extracted from the cubic Schrödinger equation (2). We consider the following decorated trees:

(59)

We observe that these trees satisfy Assumption 1 and that $ T_1 $ is a subtree of $ T_2 $ . One has that

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}( T_1) = 2 k_1^2 , \quad\mathscr{F}_{\tiny{\text{dom}}}( T_2) = 2 (k_1 + k_4)^2 \end{align*} $$

and the following quantity can be mapped back into physical space:

$$ \begin{align*} &\sum_{\substack{k= -k_1-k_4+k_2+k_3+k_5 \\ k_1\neq 0, k_1+k_4\neq 0 \\k_1,k_2,k_3,k_4,k_5\in {\mathbf{Z}}^d}} \frac{1}{ \mathscr{F}_{\tiny{\text{dom}}}( T_1)} \frac{1}{ \mathscr{F}_{\tiny{\text{dom}}}( T_2)} \overline{\hat{v}_{k_1}} \overline{\hat{v}_{k_4}}\hat{v}_{k_2}\hat{v}_{k_3}\hat{v}_{k_5} e^{i k x}\\ &\quad= \sum_{\substack{k= -k_1-k_4+k_2+k_3+k_5\\ k_1\neq 0, k_1+k_4\neq 0 \\k_1,k_2,k_3,k_4,k_5\in {\mathbf{Z}}^d}} \frac{1}{4 (k_1^2) (k_1 + k_4)^2}\overline{\hat{v}_{k_1}} \overline{\hat{v}_{k_4}}\hat{v}_{k_2}\hat{v}_{k_3}\hat{v}_{k_5} e^{i k x} \\ &\quad = \frac14 v(x)^3(-\Delta)^{-1}\left(\overline{v}(x) (-\Delta)^{-1} \overline{v}(x)\right). \end{align*} $$

2.3 Approximated decorated trees

We denote by $ {\mathcal {T}} $ the set of decorated trees $ T_{{\mathfrak {e}},r}^{{\mathfrak {n}},{\mathfrak {f}}} = (T,{\mathfrak {n}},{\mathfrak {f}},{\mathfrak {e}},r) $ where

• • $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} \in \hat {\mathcal {T}} $ .

• • The decoration of the root is given by $ r \in {\mathbf {Z}} $ , $ r \geq -1 $ such that

  (60) $$ \begin{align} r +1 \geq \deg(T_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}}) \end{align} $$
  where $ \deg $ is defined recursively by
  $$ \begin{align*} \deg({\mathbf{1}}) & = 0, \quad \deg(T_1 \cdot T_2 ) = \max(\deg(T_1),\deg(T_2)), \\ \deg({\cal I}_{({\mathfrak{t}},{\mathfrak{p}})}( \lambda^{\ell}_{k}T_1) ) & = \ell + {\mathbf{1}}_{\lbrace{\mathfrak{t}} \in {\cal L}_+\rbrace} + \deg(T_1) \end{align*} $$
  where $ {\mathbf {1}} $ is the empty forest and $ T_1, T_2 $ are forests composed of trees in $ {\mathcal {T}} $ . The quantity $\deg (T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}})$ is  the maximum number of edges with type in $ {\mathfrak {L}}_+ $ and node decorations $ {\mathfrak {n}} $ lying on the same path from one leaf to the root.

We call decorated trees in $ {\mathcal {T}} $ approximated decorated trees. The main difference with decorated trees introduced before is the adjunction of the decoration $ r $ at the root. The idea is that these trees correspond to different analytical objects. We summarise this below:

$$ \begin{align*} T_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}} & \in \hat{{\mathcal{T}}} \equiv \text{Iterated integral} \\ T_{{\mathfrak{e}},r}^{{\mathfrak{n}},{\mathfrak{f}}} & \in \hat{{\mathcal{T}}} \equiv \text{Approximation of an iterated integral of order } r. \end{align*} $$

The interpretation (83) of approximated decorated trees gives the numerical scheme; see Definition 4.4.

Example 11. We continue Example 6 with the decorated tree $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},\mathfrak {f}} $ given in (44). We suppose that $ {\mathfrak {t}}(1) $ is in $ \mathfrak {L}_+ $ but not $ {\mathfrak {t}}(2) $ and ${\mathfrak {t}}(3) $ . We are now in the context of Example 7. Then, one has

$$ \begin{align*} \deg(T_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}}) = {\mathfrak{n}}(a) + 1+ \max({\mathfrak{n}}(b),{\mathfrak{n}}(c)). \end{align*} $$

We denote by $ {\mathcal {H}} $ the vector space spanned by forests composed of trees in $ {\mathcal {T}}$ and $ \lambda ^n $ , $ n \in \mathbf {N} $ where $ \lambda ^n $ is the tree with one node decorated by $ n $ . When the decoration $ n $ is equal to zero, we identify this tree with the empty forest: $ \lambda ^{0} ={\mathbf {1}} $ . Using the symbolic notation, one has

$$ \begin{align*} {\mathcal{H}} = \langle \lbrace \prod_j \lambda^{m_j} \prod_i {\cal I}^{r_i}_{o_i}( \lambda_{k_i}^{\ell_i} F_i), \, {\cal I}_{o_i}( \lambda_{k_i}^{\ell_i} F_i) \in \hat {\mathcal{T}} \rbrace \rangle \end{align*} $$

where the product used is the forest product. We call decorated forests in $ {\mathcal {H}} $ approximated decorated forests. The map $ {\cal I}^{r}_{o}( \lambda _{k}^{\ell } \cdot ) : \hat {\mathcal {H}} \rightarrow {\mathcal {H}} $ is defined the same as for $ {\cal I}_{o}( \lambda _{k}^{\ell } \cdot ) $ except now the root is decorated by $ r $ and it could be zero if the inequality (60) is not satisfied. We extend this map to $ {\mathcal {H}} $ by

$$ \begin{align*} {\cal I}^{r}_{o}( \lambda_{k}^{\ell} (\prod_j \lambda^{m_j} \prod_i {\cal I}^{r_i}_{o_i}( \lambda_{k_i}^{\ell_i} F_i))) { \, := }{\cal I}^{r}_{o}( \lambda_{k}^{\ell+\sum_j m_j} (\prod_i {\cal I}_{o_i}( \lambda_{k_i}^{\ell_i} F_i))). \end{align*} $$

In the extension, we remove the decorations $ r_i $ and we add up the decorations $ m_j $ with $ \ell $ . In the sequel, we will use a recursive formulation and move from $ \hat {\mathcal {H}} $ to $ {\mathcal {H}} $ . Therefore, we define the map $ {\cal D}^{r} : \hat {\mathcal {H}} \rightarrow {\mathcal {H}} $ which replaces the root decoration of a decorated tree by $ r $ and performs the projection along the identity (60). It is given by

(61) $$ \begin{align} {\cal D}^{r}({\mathbf{1}})= {\mathbf{1}}_{\lbrace 0 \leq r+1\rbrace} , \quad {\cal D}^r\left( {\cal I}_{o}( \lambda_{k}^{\ell} F) \right) = {\cal I}^{r}_{o}( \lambda_{k}^{\ell} F) \end{align} $$

and we extend it multiplicatively to any forest in $ \hat {\mathcal {H}} $ . The map $ {\cal D}^r $ projects according to the order of the scheme $ r $ . We disregard decorated trees having a degree bigger than $ r $ .

We denote by $ {\mathcal {T}}_{+} $ the set of decorated forests composed of trees of the form $ (T,{\mathfrak {n}},{\mathfrak {f}},{\mathfrak {e}},(r,m)) $ where

• • $ T_{{\mathfrak {e}},r}^{{\mathfrak {n}},{\mathfrak {f}}} \in {\mathcal {T}} $ .

• • The edge connecting the root has a decoration of the form $ ({\mathfrak {t}},{\mathfrak {p}}) $ where $ {\mathfrak {t}} \in {\mathfrak {L}}_+ $ .

• • The decoration $ (r,m) $ is at the root of $ T $ and $ m \in \mathbf {N} $ is such that $ m \leq r +1 $ .

The linear span of $ {\mathcal {T}}_+ $ is denoted by $ {\mathcal {H}}_+ $ . One can observe that the main difference with ${\mathcal {H}}$ is that $ \lambda ^m \notin {\mathcal {T}}_+ $ . We can define the same grafting operator as before $ {\cal I}^{(r,m)}_{o}( \lambda _{k}^{\ell } \cdot ) : {\mathcal {H}} \rightarrow {\mathcal {H}}_+ $ the same as $ {\cal I}^{r}_{o}( \lambda _{k}^{\ell } \cdot ) : {\mathcal {H}} \rightarrow {\mathcal {H}} $ but now we add the decoration $ (r,m) $ at the root where $ m \leq r+1 $ . We also define $ \hat {\cal D}^{(r,m)}: \hat {\mathcal {H}} \rightarrow {\mathcal {H}}_+ $ the same as $ {\cal D}^{r} $ . It is given by

(62) $$ \begin{align} \hat {\cal D}^{(r,m)}({\mathbf{1}})= {\mathbf{1}}, \quad \hat {\cal D}^{(r,m)}\left( {\cal I}_{o}( \lambda_{k}^{\ell} F) \right) = {\cal I}^{(r,m)}_{o}( \lambda_{k}^{\ell} F). \end{align} $$

Example 12. For the tree (44) we obtain when applying $ {\cal D}^r $ and $ \hat {\cal D}^{(r,m)} $

(63)

For the decorated tree in (53), one obtains

(64)

2.4 Operators on approximated decorated forests

In the subsection, we introduce two maps $ \Delta $ and $ \Delta ^{\!+} $ that will act on approximated decorated forests, splitting them into two parts by the use of the tensor product. They act at two levels. First, on the shapes of the trees, they extract a subtree at the root. Then, they induce subtle changes in the decorations that could be interpreted as abstract Taylor expansions.

We define a map $\Delta : {\mathcal {H}} \rightarrow {\mathcal {H}} \otimes {\mathcal {H}}_+$ for a given $T^{{\mathfrak {n}},{\mathfrak {f}}}_{{\mathfrak {e}},r} \in {{\mathcal T}}$ by

(65) $$ \begin{align} \Delta T^{{\mathfrak{n}},{\mathfrak{f}}}_{{\mathfrak{e}},r} & = \sum_{A \in {\mathfrak{A}}(T) } \sum_{{\mathfrak{e}}_A} \frac1{{\mathfrak{e}}_A!} (A,{\mathfrak{n}} + \pi{\mathfrak{e}}_A, {\mathfrak{f}}, {\mathfrak{e}},r)\\ & \otimes \prod_{e \in \partial(A,T)}( T_{e}, {\mathfrak{n}} , {\mathfrak{f}}, {\mathfrak{e}}, (r-\deg(e),{\mathfrak{e}}_A(e))), \notag \\ & = \sum_{A \in {\mathfrak{A}}(T) } \sum_{{\mathfrak{e}}_A} \frac1{{\mathfrak{e}}_A!} A^{{\mathfrak{n}} +\pi{\mathfrak{e}}_A, {\mathfrak{f}} }_{{\mathfrak{e}},r} \otimes \prod_{e \in \partial(A,T)} (T_{e})^{{\mathfrak{n}} , {\mathfrak{f}}}_{{\mathfrak{e}}, (r-\deg(e),{\mathfrak{e}}_A(e))} \notag\end{align} $$

where we use the following notation:

• • We write $T_e $ as the planted tree above the edge $ e $ in $ T $ . For $g : E_T \rightarrow \mathbf {N}$ , we define for every $x \in N_T$ , $(\pi g)(x) = \sum _{e=(x,y) \in E_T} g(e)$ .

• • In $ A^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A, {\mathfrak {f}} }_{{\mathfrak {e}},r} $ , the maps $ {\mathfrak {n}}, {\mathfrak {f}} $ and $ {\mathfrak {e}} $ are restricted to $ N_A $ and $ E_A $ . The same is valid for $ (T_{e})^{{\mathfrak {n}} , {\mathfrak {f}}}_{{\mathfrak {e}}, (r-\deg (e),{\mathfrak {e}}_A(e))} $ where the restriction is on $ N_{T_e} \setminus \lbrace \varrho _{T_e}\rbrace $ and $ E_{T_e} $ , $ \varrho _{T_e} $ is the root of $ T_e $ . When $ A $ is reduced to a single node, we set $ A^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A, {\mathfrak {f}} }_{{\mathfrak {e}},r} = \lambda ^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A} $ .

• • The first sum runs over ${\mathfrak {A}}(T)$ , the set of all subtrees A of T containing the root $ \varrho $ of $ T $ . The second sum runs ${\mathfrak {e}}_{A} : \partial (A,T) \rightarrow \mathbf {N}$ where $\partial (A,T)$ denotes the edges in $E_T \setminus E_A$ of type in $ {\mathfrak {L}}_+ $ that are adjacent to $N_A$ .

• • Factorial coefficients are understood in multi-index notation.

• • We define $ \deg (e) $ for $ e \in E_T $ as the number of edges having $ {\mathfrak {t}}(e) \in {\mathfrak {L}}_+ $ lying on the path from $ e $ to the root in the decorated tree $ T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}} $ . We also add up the decoration $ {\mathfrak {n}} $ on this path.

Let us comment briefly that the play on decorations can be interpreted as asbtract Taylor expansions. We can use the following dictionary:

$$ \begin{align*} (T_e)_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}} \equiv \int_{0}^{\tau} e^{i\xi P(k)} f(k_1,\ldots ,k_n,\xi) d\xi \end{align*} $$

where $ k_1,\ldots ,k_n $ are the frequencies appearing on the leaves of $ (T_e)_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}}$ , $ P $ is the polynomial associated to the decoration of the edge $ e $ and $ k $ is the frequency on the node connected to the root. This iterated integral appears inside the iterated integral associated to $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} $ . For our numerical approximation, we need to approximate this integral by giving a scheme of the form

$$ \begin{align*} (T_e)_{{\mathfrak{e}}, \tilde{r}}^{{\mathfrak{n}},{\mathfrak{f}}} \equiv \sum_{\ell \leq \tilde{r}} \frac{\tau^{\ell}}{\ell!} f_{\tilde{r},\ell}(k_1,\ldots ,k_n), \quad (T_{e})^{{\mathfrak{n}} , {\mathfrak{f}}}_{{\mathfrak{e}}, (\tilde{r},\ell)} \equiv f_{\tilde{r},\ell}(k_1,\ldots ,k_n) \end{align*} $$

where the order of the expansion is given by $ \tilde {r} = r - \deg (e) $ . Then, the $ \tau ^{\ell } $ are part of the original iterated integrals and cannot be detached as the $ f_{\tilde {r},\ell } $ . That is why we increase the polynomial decorations where the tree $ T_e $ was originally attached via the term $ {\mathfrak {n}} +\pi {\mathfrak {e}}_A $ . The choice for $ \tilde {r} $ is motivated by the fact that we do not need to go too far for the approximation of $ (T_e)_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} $ . We take into account all of the time integrals and polynomials which lie on the path connecting the root of $ T_e $ to the root of $ T $ .

The map $ \Delta $ is compatible with the projection induced by the decoration $ r $ . Indeed, one has

$$ \begin{align*} \deg(T_{{\mathfrak{e}}}^{{\mathfrak{n}}, {\mathfrak{f}}}) = \max_{e \in E_T} \left( \deg( (T_e)_{{\mathfrak{e}}}^{{\mathfrak{n}}, {\mathfrak{f}}} ) + \deg(e) \right). \end{align*} $$

Therefore, if $ \deg (T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}}) < r $ , then for every $ e \in E_T $

$$ \begin{align*} \deg( (T_e)_{{\mathfrak{e}}}^{{\mathfrak{n}}, {\mathfrak{f}}} ) + \deg(e) < r. \end{align*} $$

We deduce that if $ T^{{\mathfrak {n}},{\mathfrak {f}}}_{{\mathfrak {e}},r} $ is zero, then the $ (T_e)_{{\mathfrak {e}},(r-\deg (e),{\mathfrak {e}}_A(e))}^{{\mathfrak {n}}, {\mathfrak {f}}} $ are zero, too.

We define a map ${\Delta ^{\!+}} : {\mathcal {H}}_+ \rightarrow {\mathcal {H}}_+ \otimes {\mathcal {H}}_+$ given for $T^{{\mathfrak {n}}, {\mathfrak {f}}}_{{\mathfrak {e}},(r,m)} \in {{\mathcal T}}_+$ by

(66) $$ \begin{align} {\Delta^{\!+}} T^{{\mathfrak{n}},{\mathfrak{f}}}_{{\mathfrak{e}},(r,m)} = \sum_{A \in {\mathfrak{A}}(T) } \sum_{{\mathfrak{e}}_A} \frac1{{\mathfrak{e}}_A!} A^{{\mathfrak{n}} +\pi{\mathfrak{e}}_A, {\mathfrak{f}} }_{{\mathfrak{e}},(r,m)} \otimes \prod_{e \in \partial(A,T)} (T_{e})^{{\mathfrak{n}} , {\mathfrak{f}}}_{{\mathfrak{e}}, (r-\deg(e),{\mathfrak{e}}_A(e))}. \end{align} $$

We require that $ A^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A, {\mathfrak {f}} }_{{\mathfrak {e}},(r,m)} \in {\mathcal {H}}_+ $ , so we implicitly have a projection on zero when $ A $ happens to be a single node with $ {\mathfrak {n}} +\pi {\mathfrak {e}}_A \neq 0 $ . We illustrate this coproduct on a well-chosen example.

Example 13. We continue with the tree in Example 5. We suppose that $ {\mathfrak {L}}_+ = \lbrace {\mathfrak {t}}(2), {\mathfrak {t}}(3), {\mathfrak {t}}(4), {\mathfrak {t}}(5) \rbrace $ . Below, the subtree $ A \in {\mathfrak {A}}(T)$ is coloured in blue. We have $ N_A = \lbrace \varrho , a,b\rbrace $ , $ E_A = \lbrace 1,2 \rbrace $ and $ \partial (A,T) = \lbrace 3, 4,5 \rbrace $ .

We also get

$$ \begin{align*} \deg(4) = \deg(5) = 1 + {\mathfrak{n}}(a) + {\mathfrak{n}}(b), \quad \deg(3) = {\mathfrak{n}}(a). \end{align*} $$

We have for a fixed $ {\mathfrak {e}}_A : \partial (A,T) \rightarrow \mathbf {N} $ :

where $ \bar r = r - {\mathfrak {n}}(a) - {\mathfrak {n}}(b) -1 $ . Now if $ A $ is just equal to the root of $ T $ , then one gets $ N_A = \lbrace \varrho \rbrace $ , $ E_A = \emptyset $ and $ \partial (A,T) = \lbrace 1\rbrace $ as illustrated below:

(67)

The map $ {\Delta ^{\!+}} $ behaves the same way except that we start with a tree decorated by $ (r,m) $ at the root and we exclude the case described in (67).

Example 14. Next we provide a more explicit example of the computations for the maps $ {\Delta ^{\!+}} $ and $ \Delta $ on the tree

(68)

that appears for the KdV equation (3). We have that

where $ \ell = k_1 + k_2 $ and we have used similar notations as in Example 7 and in (64). We have introduced a new graphical notation for a node decorated by the decoration $(m,\ell )$ when $ m \neq 0 $ :

. One can notice that the second abstract Taylor expansion is shorter. This is due to the fact that there was one blue edge (in $ {\mathfrak {L}}_+ $ ) on the path connecting the cutting tree to the root. In addition, we have

One of the main differences between $ \Delta $ and $ {\Delta ^{\!+}} $ is that we do not have a higher Taylor expansion on the edge connecting the root for $ {\Delta ^{\!+}} $ because the elements $ \lambda ^n $ are not in $ {\mathcal {H}}_+ $ . With this property, $ {\mathcal {H}}_+ $ will be a connected Hopf algebra.

We use the symbolic notation to provide an alternative, recursive definition of the two maps $ \Delta : {\mathcal {H}} \rightarrow {\mathcal {H}} \otimes {\mathcal {H}}_+ $ and $ {\Delta ^{\!+}} : {\mathcal {H}}_+ \rightarrow {\mathcal {H}}_+ \otimes {\mathcal {H}}_+ $ :

(69) $$ \begin{align} \begin{split} \Delta {\mathbf{1}} & = {\mathbf{1}} \otimes {\mathbf{1}}, \quad \Delta \lambda^{\ell} = \lambda^{\ell} \otimes {\mathbf{1}}\\ \Delta {\cal I}^{r}_{o_1}( \lambda_{k}^{\ell} F) & = \left( {\cal I}^{r}_{o_1}( \lambda_{k}^{\ell}\cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell}(F) \\ \Delta {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} F) & = \left( {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F) + \sum_{m \leq r +1} \frac{ \lambda^{m}}{m!} \otimes {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) \\ {\Delta^{\!+}} {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) & = \left( {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F) + {\mathbf{1}} \otimes {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) \end{split} \end{align} $$

where $ o_1 = ({\mathfrak {t}}_1,p_1) $ , $ {\mathfrak {t}}_1 \notin {\mathfrak {L}}_+ $ and $ o_2 = ({\mathfrak {t}}_2,p_2) $ , $ {\mathfrak {t}}_2 \in {\mathfrak {L}}_+ $ .

In the next proposition, we prove the equivalence between the recursive and nonrecursive definitions.

Proof

The operator $\Delta $ is multiplicative on $\hat {\mathcal {H}}$ . It remains to verify that the recursive identities hold as well. We consider $\Delta \sigma $ with $ \sigma = {\cal I}^{r}_{({\mathfrak {t}}_2,p)}( \lambda ^{\ell }_{k} \tau )$ and $ {\mathfrak {t}}_2 \in {\mathfrak {L}}_+ $ . We write $ \tau = F_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}}$ , $ \bar \tau = F_{{\mathfrak {e}}, r-\ell -1}^{{\mathfrak {n}},{\mathfrak {f}}} $ where $ F = \prod _i T_i $ is a forest formed of the trees $ T_i $ . One has $ \bar \tau = \prod _i (T_i)_{{\mathfrak {e}}_i, r-\ell -1}^{{\mathfrak {n}}_i,{\mathfrak {f}}_i} $ and the maps $ {\mathfrak {n}}, {\mathfrak {f}}, {\mathfrak {e}} $ are obtained as disjoint sums of the $ {\mathfrak {n}}_i, {\mathfrak {f}}_i, {\mathfrak {e}}_i $ . We write $\sigma = T^{\bar {\mathfrak {n}},\bar {\mathfrak {f}}}_{\bar {\mathfrak {e}},r}$ where

$$ \begin{align*} \bar {\mathfrak{e}} = {\mathfrak{e}} + {\mathbf{1}}_e ({\mathfrak{t}}_2,p), \quad \bar {\mathfrak{f}}(u) = {\mathfrak{f}} + {\mathbf{1}}_u k, \quad \bar {\mathfrak{n}}(u) = {\mathfrak{n}} + {\mathbf{1}}_u \ell \end{align*} $$

and e denotes a trunk of type ${\mathfrak {t}}$ created by ${\cal I}_{({\mathfrak {t}}_2,p)}$ , $\rho $ is the root of T and $ u $ is such that $ e=(\rho ,u) $ . It follows from these definitions that

$$ \begin{align*} {\mathfrak{A}}(T) = \{\{\rho\}\} \cup \{ A \cup \{\rho,e\}\,:\, A \in {\mathfrak{A}}(F)\}\; \end{align*} $$

where $ {\mathfrak {A}}(F) = \sqcup _i {\mathfrak {A}}(T_i)$ and the $ A $ are forests. One can actually rewrite (65) exactly the same for forests. Then, we have the identity

$$ \begin{align*} \Delta \sigma &= ({\cal I}^{r}_{({\mathfrak{t}}_2,p)}( \lambda^{\ell}_k \cdot) \otimes {\mathrm{id}}) \Delta \bar \tau + \sum_{{\mathfrak{e}}_{\bullet}} {1\over {\mathfrak{e}}_{\bullet}!} (\bullet, \pi {\mathfrak{e}}_{\bullet},0,0 ,0) \otimes (T, \bar{{\mathfrak{n}}},\bar{{\mathfrak{f}}},\bar{{\mathfrak{e}}},(r,{\mathfrak{e}}_{\bullet})) \\[-2pt] & = \left( {\cal I}^{r}_{({\mathfrak{t}}_2,p)}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(\tau) + \sum_{m \leq r +1} \frac{ \lambda^{m}}{m!} \otimes {\cal I}^{(r,m)}_{({\mathfrak{t}}_2,p)}( \lambda_{k}^{\ell} \tau)\; \end{align*} $$

where the recursive $\left ( {\cal I}^{r}_{({\mathfrak {t}}_2,p)}( \lambda ^{\ell }_k \cdot ) \otimes {\mathrm {id}} \right ) \Delta $ encodes the extraction of $ A \cup \{\rho ,e\} $ , $ A \in {\mathfrak {A}}(T) $ . We can perform a similar proof for $ {\mathfrak {t}}_1 \notin {\mathfrak {L}}_+ $ and for $ {\Delta ^{\!+}} $ . The main difference is that the sum on the polynomial decoration is removed for an edge not in $ {\mathfrak {L}}_+ $ and for $ {\Delta ^{\!+}} $ such that we just keep the first term.

Example 15. We illustrate the recursive definition of $ \Delta $ by performing some computations on some relevant decorated trees that one can face in practice; for instance, in case of cubic NLS (2). For the decorated tree

$$ \begin{align*} T_1 = {\cal I}_{({\mathfrak{t}}_2,0)} \left( \lambda_k F_1 \right) \quad F_1 = {\cal I}_{({\mathfrak{t}}_1,1)}( \lambda_{k_1}) {\cal I}_{({\mathfrak{t}}_1,0)}( \lambda_{k_2}) {\cal I}_{({\mathfrak{t}}_1,0)}( \lambda_{k_3}) \end{align*} $$

which appears in context of the cubic Schrödinger equation (see Subsection 5.1), we have that

(70) $$ \begin{align} \Delta {\cal D}^r(T_1) = {\cal D}^r(T_1) \otimes {\mathbf{1}} + \sum_{m \leq r+1} \frac{ \lambda^m}{m!} \otimes \hat {\cal D}^{(r,m)}( T_1). \end{align} $$

Relation (70) is proven as follows: Using the definition of $ {\cal D}^r$ in (61) as well as (69) yields that

(71) $$ \begin{align} \Delta {\cal D}^r(T_1) & = \Delta {\cal I}^{r}_{({\mathfrak{t}}_2,0)}( \lambda_{k} F_1)\\ & = \left( {\cal I}^{r}_{({\mathfrak{t}}_2,0)}( \lambda_k \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-1}(F_1) + \sum_{m \leq r +1} \frac{ \lambda^{m}}{m!} \otimes {\cal I}^{(r,m)}_{({\mathfrak{t}}_2,0)}( \lambda_k F_1). \notag \end{align} $$

Thanks to the definition of $\hat {\cal D}^r$ in (62), we can conclude that

$$ \begin{align*} {\cal I}^{(r,m)}_{({\mathfrak{t}}_2,0)}( \lambda_k F_1) = \hat {\cal D}^{(r,m)}{\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{k} F_1) =\hat {\cal D}^{(r,m)} (T_1), \end{align*} $$

which yields together with (71) that

(72) $$ \begin{align} \Delta {\cal D}^r(T_1) = \left( {\cal I}^{r}_{({\mathfrak{t}}_2,0)}( \lambda_k \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-1}(F_1) + \sum_{\ell \leq r +1} \frac{ \lambda^{\ell}}{\ell!} \otimes \hat {\cal D}^{(r,\ell)} (T_1). \end{align} $$

Next we need to analyse the term $ \left ( {\cal I}^{r}_{({\mathfrak {t}}_2,0)}( \lambda _k \cdot ) \otimes {\mathrm {id}} \right ) \Delta {\cal D}^{r-1}(F_1) $ . First, we use the multiplicativity of ${\cal D}^r$ , (cf. (61)) and coproduct which yields that

(73) $$ \begin{align} \Delta {\cal D}^{r-1}(F_1) = \left( \Delta {\cal I}^{r-1}_{({\mathfrak{t}}_1,1)}( \lambda_{k_1})\right)\left( \Delta {\cal I}^{r-1}_{({\mathfrak{t}}_1,0)}( \lambda_{k_2}) \right)\left(\Delta {\cal I}^{r-1}_{({\mathfrak{t}}_1,0)}( \lambda_{k_3})\right). \end{align} $$

Thanks to (69), we furthermore have that

(74) $$ \begin{align} \Delta {\cal I}^{r-1}_{({\mathfrak{t}}_1,p)}( \lambda_{k_j}) &= \left( {\cal I}^{r-1}_{({\mathfrak{t}}_1,p)}( \lambda_{k_j}\cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-1}( {\mathbf{1}})\\ & = \left( {\cal I}^{r-1}_{({\mathfrak{t}}_1,p)}( \lambda_{k_j}\cdot) \otimes {\mathrm{id}} \right)\left( {\mathbf{1}} \otimes {\mathbf{1}}\right) = {\cal I}^{r-1}_{({\mathfrak{t}}_1,p)}( \lambda_{k_j}) \otimes {\mathbf{1}} \notag\end{align} $$

where we have used that $ {\cal D}^{r-1}( {\mathbf {1}} ) = {\mathbf {1}}$ and $\Delta {\mathbf {1}} = {\mathbf {1}} \otimes {\mathbf {1}}$ ; see also (69). Plugging (74) into (73) yields that

(75) $$ \begin{align} &\Delta {\cal D}^{r-1}(F_1)\\ & \quad = \left( {\cal I}^{r-1}_{({\mathfrak{t}}_1,1)}( \lambda_{k_1}) \otimes {\mathbf{1}} \right) \left( {\cal I}^{r-1}_{({\mathfrak{t}}_1,0)}( \lambda_{k_2}) \otimes {\mathbf{1}} \right) \left( {\cal I}^{r-1}_{({\mathfrak{t}}_1,0)}( \lambda_{k_3}) \otimes {\mathbf{1}} \right)\notag \\ & \quad = {\cal I}^{r-1}_{({\mathfrak{t}}_1,1)}( \lambda_{k_1}) {\cal I}^{r-1}_{({\mathfrak{t}}_1,0)}( \lambda_{k_2}) {\cal I}^{r-1}_{({\mathfrak{t}}_1,0)}( \lambda_{k_3}) \otimes {\mathbf{1}} = {\cal D}^{r-1}(F_1) \otimes {\mathbf{1}}. \notag \end{align} $$

Hence,

$$ \begin{align*} \left( {\cal I}^{r}_{({\mathfrak{t}}_2,0)}( \lambda_k \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-1}(F_1)& = \left( {\cal I}^{r}_{({\mathfrak{t}}_2,0)}( \lambda_k \cdot) \otimes {\mathrm{id}} \right) \left( {\cal D}^{r-1}(F_1) \otimes {\mathbf{1}} \right)\\ & = {\cal I}^{r}_{({\mathfrak{t}}_2,0)}( \lambda_k F_1) \otimes {\mathbf{1}} = {\cal D}^{r}(T_1) \otimes {\mathbf{1}}. \end{align*} $$

Plugging this into (72) yields (70).

2.5 Hopf algebra and comodule structures

Using the two maps $ \Delta $ and $ {\Delta ^{\!+}} $ , we want to identify a comodule structure over a Hopf algebra. Here, we provide a brief reminder of this structure for a reader not familiar with it. For simplicity, we will use the notation of the spaces introduced above as well as the maps $ \Delta $ and $ {\Delta ^{\!+}} $ . The proof that we are indeed in this framework is then given in Proposition 2.15.

A bialgebra $({\mathcal {H}}_+,{\mathcal {M}},{\mathbf {1}},{\Delta ^{\!+}},{\mathbf {1}}^\star )$ is given by

• • A vector space $ {\mathcal {H}}_+ $ over $\mathbf {C}$

• • A linear map ${\mathcal {M}}:{\mathcal {H}}_+\otimes {\mathcal {H}}_+ \to {\mathcal {H}}_+$ (product) and an element $\eta :r\mapsto r{\mathbf {1}}$ , ${\mathbf {1}}\in {\mathcal {H}}_+$ (identity) such that $({\mathcal {H}}_+,{\mathcal {M}},\eta )$ is a unital associative algebra.

• • Linear maps ${\Delta ^{\!+}} :{\mathcal {H}}_+ \to {\mathcal {H}}_+\otimes {\mathcal {H}}_+$ (coproduct) and ${\mathbf {1}}^\star :{\mathcal {H}}_+\to \mathbf {C}$ (counit), such that $({\mathcal {H}}_+,{\Delta ^{\!+}},{\mathbf {1}}^\star )$ is a counital coassociative coalgebra, namely,

  (76) $$ \begin{align} ({\Delta^{\!+}}\otimes{\mathrm{id}}){\Delta^{\!+}}=({\mathrm{id}}\otimes{\Delta^{\!+}}){\Delta^{\!+}}, \qquad ({\mathbf{1}}^\star\otimes{\mathrm{id}}){\Delta^{\!+}}= ({\mathrm{id}}\otimes{\mathbf{1}}^\star){\Delta^{\!+}}={\mathrm{id}}. \end{align} $$

• • $ {\Delta ^{\!+}} $ and $ {\mathbf {1}}^{*} $ (respectively $ {\mathcal {M}} $ and $ {\mathbf {1}} $ ) are homomorphisms of algebras (coalgebras).

A Hopf algebra is a bialgebra $({\mathcal {H}}_+,{\mathcal {M}},{\mathbf {1}},{\Delta ^{\!+}},{\mathbf {1}}^\star )$ endowed with a linear map ${\mathcal {A}}: {\mathcal {H}}_+ \to {\mathcal {H}}_+$ such that

(77) $$ \begin{align} {\mathcal{M}}({\mathrm{id}}\otimes {\mathcal{A}}){\Delta^{\!+}} = {\mathcal{M}}({\mathcal{A}}\otimes{\mathrm{id}}){\Delta^{\!+}}= {\mathbf{1}}^\star{\mathbf{1}}. \end{align} $$

A right comodule over a bialgebra $({\mathcal {H}}_+,{\mathcal {M}},{\mathbf {1}},{\Delta ^{\!+}},{\mathbf {1}}^\star )$ is a pair $({\mathcal {H}},\Delta )$ where ${\mathcal {H}}$ is a vector space and $\Delta : {\mathcal {H}} \to {\mathcal {H}} \otimes {\mathcal {H}}_+$ is a linear map such that

(78) $$ \begin{align} (\Delta\otimes{\mathrm{id}})\Delta=({\mathrm{id}}\otimes{\Delta^{\!+}})\Delta, \qquad ({\mathrm{id}} \otimes {\mathbf{1}}^{\star})\Delta={\mathrm{id}}. \end{align} $$

In our framework, the product $ {\mathcal {M}} $ is given by the forest product

$$ \begin{align*} {\mathcal{M}} (F_1 \otimes F_2) = F_1 \cdot F_2. \end{align*} $$

Most of the properties listed above are quite straightforward to check. In the next proposition we focus on the coassociativity of the maps $ \Delta $ and $ {\Delta ^{\!+}} $ in (76) and (78). First, let us explain why this structure is useful. If one considers characters that are multiplicative maps $ g : {\mathcal {H}}_+ \rightarrow \mathbf {C}$ , then the coproduct $ \Delta ^{\!+} $ and the antipode ${\mathcal {A}} $ allow us to put a group structure on them. We denote the group of such characters by $ \mathcal {G} $ . The product for this group is the convolution product $ \star $ given for $ f,g \in \mathcal {G} $ by

$$ \begin{align*} f \star g = \left( f \otimes g \right) \Delta^{\!+}. \end{align*} $$

We do not need a multiplication because we use the identification $ \mathbf {C} \otimes \mathbf {C} \cong \mathbf {C} $ . The inverse is given by the antipode

$$ \begin{align*} f^{-1} = f({\mathcal{A}} \cdot). \end{align*} $$

The comodule structure allows us to have an action of $ \mathcal {G} $ onto $ {\mathcal {H}} $ defined by

$$ \begin{align*} \Gamma_{f} = \left( {\mathrm{id}} \otimes f \right) \Delta^{\!+}, \quad \Gamma_{f} \Gamma_{g} = \Gamma_{f \star g}, \quad \Gamma_f^{-1} = \Gamma_{f^{-1}}. \end{align*} $$

In Subsection 3.2, we will use these structures to decompose our scheme for iterated integrals – that is, a character $ \Pi ^n : {\mathcal {H}} \rightarrow {{\cal C}}$ – into

$$ \begin{align*} \Pi^n = \left( \hat \Pi^n \otimes A^n \right) \Delta \end{align*} $$

where $ {{\cal C}} $ is a space introduced in Subsection 3.1, $ A^n \in \mathcal {G} $ (defined from $ \Pi ^n $ ) and $ \hat \Pi ^n : {\mathcal {H}} \rightarrow {{\cal C}}$ . In the identity above we have made the following identification $ {{\cal C}} \otimes \mathbf {C} \cong {{\cal C}} $ . The main point of this decomposition is to let the character $ \hat \Pi ^n $ appear, which is simpler than $ \Pi _n $ . This will help us in carrying out the local error analysis in Subsection 3.3.

Proposition 2.14. One has

$$ \begin{align*} \left( \Delta \otimes {\mathrm{id}} \right) \Delta = \left( {\mathrm{id}} \otimes {\Delta^{\!+}} \right) \Delta, \quad \left( {\Delta^{\!+}} \otimes {\mathrm{id}} \right) {\Delta^{\!+}} = \left( {\mathrm{id}} \otimes {\Delta^{\!+}} \right) {\Delta^{\!+}}. \end{align*} $$

Proof

We proceed by induction and we perform the proof only for $ {\cal I}^{r}_{o_2}( \lambda _{k}^{\ell } F) $ . The other case follows similar steps. Note that

$$ \begin{align*} & \left( \Delta \otimes {\mathrm{id}} \right) \Delta {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} F) \\ & = \left(\Delta {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F) + \sum_{m \leq r + 1} \Delta \frac{ \lambda^{m}}{m!} \otimes {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) \\ & = \sum_{m \leq r + 1} \frac{ \lambda^{m}}{m!} \otimes \left( {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F) \\ & \quad{}+ \sum_{m \leq r +1} \frac{ \lambda^{m}}{m!} \otimes {\mathbf{1}} \otimes {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) +\left( \left( {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \right) \Delta \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F). \end{align*} $$

On the other hand, we get

$$ \begin{align*} & \left( {\mathrm{id}} \otimes {\Delta^{\!+}} \right) \Delta {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} F) \\ & = \left({\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\Delta^{\!+}} \right) \Delta {\cal D}^{r-\ell-1}(F) + \sum_{m \leq r+1} \frac{ \lambda^{m}}{m!} \otimes {\Delta^{\!+}} {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) \\ & = \left({\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\Delta^{\!+}} \right) \Delta {\cal D}^{r-\ell-1}(F) + \sum_{m \leq r +1} \frac{ \lambda^{m}}{m!} \otimes {\mathbf{1}} \otimes {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) \\ & \quad{}+\sum_{m \leq r +1} \frac{ \lambda^{m}}{m!} \otimes \left( {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F). \end{align*} $$

Next we observe that

$$ \begin{align*} \left({\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\Delta^{\!+}} \right) \Delta {\cal D}^{r-\ell-1}(F) & = \left( {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \otimes {\mathrm{id}} \right) \left( {\mathrm{id}} \otimes {\Delta^{\!+}} \right) \Delta {\cal D}^{r-\ell-1}(F) \\ & = \left( {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \otimes {\mathrm{id}} \right) \left( \Delta \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F) \\ & = \left( \left( {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \right) \Delta \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F) \end{align*} $$

where we used an inductive argument for

$$ \begin{align*} \left( \Delta \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F) = \left( {\mathrm{id}} \otimes {\Delta^{\!+}} \right) \Delta {\cal D}^{r-\ell-1}(F). \end{align*} $$

This yields the assertion.

Proof

From Proposition 2.14, $ {\mathcal {H}}_+ $ is a bialgebra and $ \Delta $ is a coaction. In fact, $ {\mathcal {H}}_+ $ is a connected graded bialgebra with the grading given by the number of edges. Therefore, from [Reference Manchon66, Corollary II.3.2], it is a Hopf algebra and we get the existence of a unique map called the antipode such that

(79) $$ \begin{align} {\mathcal{M}} \left( {\mathcal{A}} \otimes {\mathrm{id}} \right) {\Delta^{\!+}} = {\mathcal{M}} \left( {\mathrm{id}} \otimes {\mathcal{A}} \right) {\Delta^{\!+}} = {\mathbf{1}} {\mathbf{1}}^{\star}. \end{align} $$

This concludes the proof.

We use the identity (79) to write a recursive formulation for the antipode.

Proposition 2.16. For every $ T \in \hat {\mathcal {H}} $ , one has

(80) $$ \begin{align} {\mathcal{A}} {\cal I}^{(r,m)}_{o_2}( \lambda_k^{\ell} F ) = {\mathcal{M}} \left( {\cal I}^{(r,m)}_{o_2}( \lambda_k^{\ell} \cdot )\otimes {\mathcal{A}} \right) \Delta {\cal D}^{r-\ell-1}(F). \end{align} $$

Proof

We use the identity (79), which implies that

$$ \begin{align*} {\mathcal{M}} \left( {\mathrm{id}} \otimes {\mathcal{A}} \right) {\Delta^{\!+}} {\cal I}^{(r,m)}_{o_2}( \lambda_k^{\ell} F )= {\mathbf{1}} \, {\mathbf{1}}^{\star}\left( {\cal I}^{(r,m)}_{o_2}( \lambda_k^{\ell} F ) \right). \end{align*} $$

As ${\mathbf {1}}^{*}$ is nonzero only on the empty forest, we can thus conclude that

$$ \begin{align*} {\mathcal{M}} \left( {\mathrm{id}} \otimes {\mathcal{A}} \right) {\Delta^{\!+}} {\cal I}^{(r,m)}_{o_2}( \lambda_k^{\ell} F )= 0.\end{align*} $$

Then, we have by the definition of ${\Delta ^{\!+}} {\cal I}^{(r,m)}_{o_2}( \lambda _k^{\ell } F )$ given in (69) that

$$ \begin{align*} {\mathcal{M}} \left( {\mathrm{id}} \otimes {\mathcal{A}} \right) \left( \left( {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} \cdot) \otimes {\mathrm{id}} \right) \Delta {\cal D}^{r-\ell-1}(F) + {\mathbf{1}} \otimes {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) \right)= 0, \end{align*} $$

which yields (80).

Remark 2.17. The formula (80) can be rewritten in a nonrecursive form. Indeed, let us introduce the reduced coproduct:

$$ \begin{align*} \tilde{\Delta} F = {\Delta^{\!+}} F - F \otimes {\mathbf{1}} - {\mathbf{1}} \otimes F. \end{align*} $$

Then, we can rewrite (79) as follows:

(81) $$ \begin{align} {\mathcal{A}} F = - F - \sum_{(T)} F' \cdot ({\mathcal{A}} F'') \quad \tilde \Delta F = \sum_{(T)} F' \otimes F'', \end{align} $$

where we have used Sweedler notations.

Example 16. We compute the antipode on some KdV trees (cf. (51) and (68)) using the formula (81)

3.1 A recursive formulation

For the rest of this section, an element of $ {\mathfrak {L}}_+ $ (respectively $ {\mathfrak {L}}_+ \times \lbrace 0,1\rbrace $ ) is denoted by $ {\mathfrak {t}}_2 $ (respectively $ o_2 $ ) and an element of $ {\mathfrak {L}} \setminus {\mathfrak {L}}_+ $ (respectively $ {\mathfrak {L}} \setminus {\mathfrak {L}}_+ \times \lbrace 0,1 \rbrace $ ) is denoted by $ {\mathfrak {t}}_1 $ (respectively $ o_1 $ ). We denote by $ {{\cal C}} $ the space of functions of the form $ z \mapsto \sum _j Q_j(z)e^{i z P_j(k_1,\ldots ,k_n) } $ where the $ Q_j(z) $ are polynomials in $ z $ and the $ P_j $ are polynomials in $ k_1,\ldots ,k_n \in {\mathbf {Z}}^{d} $ . The $ Q_j $ may also depend on $ k_1,\ldots ,k_n $ . We use the pointwise product on $ {{\cal C}} $ for $ G_1(z) = Q_1(z)e^{i z P_1(k_1,\ldots ,k_n) } $ and $ G_2(z) = Q_2(z)e^{i z P_2( k_1,\ldots , k_n) } $ given by

$$ \begin{align*} ( G_1 G_2)(z) = Q_1(z) Q_2(z) e^{i z P(k_1,\ldots ,k_n)}, \quad P = P_1 + P_2. \end{align*} $$

We want to define characters on decorated trees using their recursive construction. A character is a map defined from $ \hat {\mathcal {H}} $ into $ {{\cal C}} $ which respects the forest product. In the sense that $ g : \hat {\mathcal {H}} \rightarrow {{\cal C}} $ is a character if one has

$$ \begin{align*} g(F \cdot \bar F) = g(F) g(\bar F), \quad F, \bar{F} \in \hat {\mathcal{H}}. \end{align*} $$

We define the following character $ \Pi : \hat {\mathcal {H}} \rightarrow \mathcal {C} $ by

(82) $$ \begin{align} \begin{split} \Pi \left( F \cdot \bar F \right)(\tau) & = ( \Pi F)(\tau) ( \Pi \bar F )(\tau), \\[-1pt] \Pi \left( {\cal I}_{o_1}( \lambda_k^{\ell} F)\right)(\tau) & = e^{i \tau P_{o_1}(k)} \tau^{\ell} (\Pi F)(\tau), \\[-1pt] \Pi \left( {\cal I}_{o_2}( \lambda_k^{\ell} F)\right)(\tau) & = -i \vert \nabla\vert^{\alpha} (k) \int_{0}^{\tau} e^{i \xi P_{o_2}(k)} \xi^{\ell}(\Pi F)(\xi) d \xi, \end{split} \end{align} $$

where $ F, \bar F \in \hat {\mathcal {H}} $ .

Example 17. With the aid of (82), one can compute recursively the following oscillatory integrals arising in the cubic NLS equation (2):

We need a well-chosen approximation of order $ r $ for the character $ \Pi $ defined in (82), which is suitable in the sense that it embeds those dominant frequencies matching the regularity n of the solution; see Remark 1.1. Therefore, we consider a new family of characters defined now on $ {\mathcal {H}} $ and parametrised by $ n \in \mathbf {N} $ :

(83) $$ \begin{align} \begin{split} \Pi^n \left( F \cdot \bar F \right)(\tau) & = \left( \Pi^n F \right)(\tau) \left( \Pi^n \bar F \right)(\tau), \quad (\Pi^n \lambda^{\ell})(\tau) = \tau^{\ell}, \\ (\Pi^n {\cal I}^{r}_{o_1}( \lambda_{k}^{\ell} F ))(\tau) & =\tau^{\ell} e^{i \tau P_{o_1}(k)} (\Pi^n {\cal D}^{r-\ell}(F))(\tau), \\ \left( \Pi^n {\cal I}^{r}_{o_2}( \lambda^{\ell}_k F) \right) (\tau) & = {{\cal K}}^{k,r}_{o_2} \left( \Pi^n \left( \lambda^{\ell} {\cal D}^{r-\ell-1}(F) \right),n \right)(\tau). \end{split} \end{align} $$

All approximations are thereby carried out in the map $ {{\cal K}}^{k,r}_{o_2} \left ( \cdot ,n \right ) $ which is given in Definition 3.1. The main idea behind the map $ {{\cal K}}^{k,r}_{o_2} \left ( \cdot ,n \right ) $ is that all integrals are approximated through well-chosen Taylor expansions depending on the regularity $ n $ of the solution assumed a priori and the interaction of the frequencies in the decorated trees. For a polynomial $ P(k_1,\ldots ,k_n)$ , we define the degree of $ P $ denoted by $ \deg (P) $ as the maximum $ m $ such that $ k_i^{m} $ appears as a factor of one monomial in $ P $ for some i. For example:

$$ \begin{align*} & P(k_1,k_2,k_3) = - 2 k_1 (k_2+k_3) + 2 k_2 k_3, \quad \deg(P) = 1,\\ & P(k_1,k_2,k_3) = k_1^2- 2 k_1 (k_2+k_3) + 2 k_2 k_3 , \quad \deg(P) = 2. \end{align*} $$

Definition 3.1. Assume that $ G: \xi \mapsto \xi ^{q} e^{i \xi P(k_1,\ldots ,k_n)} $ where $ P $ is a polynomial in the frequencies $ k_1,\ldots ,k_n $ and let $ o_2 = ({\mathfrak {t}}_2,p) \in {\mathfrak {L}}_+ \times \lbrace 0,1 \rbrace $ and $ r \in \mathbf {N} $ . Let $ k $ be a linear map in $ k_1,\ldots ,k_n $ using coefficients in $ \lbrace -1,0,1 \rbrace $ and

$$ \begin{align*} \begin{split} {\cal L}_{\tiny{\text{dom}}} & = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{o_2}(k) + P \right), \quad {\cal L}_{\tiny{\text{low}}} = {\mathcal{P}}_{\tiny{\text{low}}} \left( P_{o_2}(k) + P \right) \\ f(\xi) & = e^{i \xi {\cal L}_{\tiny{\text{dom}}}}, \quad g(\xi) = e^{i \xi {\cal L}_{\tiny{\text{low}}}}, \quad { \tilde g}(\xi) = e^{i \xi \left( P_{o_2}(k) + P \right)}. \end{split} \end{align*} $$

Then, we define for $ n \in \mathbf {N} $ and $ r \geq q $

(84) $$ \begin{align} {{\cal K}}^{k,r}_{o_2} ( { G},n)(\tau) = \left\{ \begin{array}{l} \displaystyle -i \vert \nabla\vert^\alpha(k) \sum_{\ell \leq r - q} \frac{ { \tilde g}^{(\ell)}(0)}{\ell!} \int_0^{\tau} \xi^{\ell+q} d \xi, \, \text{if } n \geq \text{deg}\left(\mathcal{L}_{\tiny{\text{dom}}}^{r+1}\right) + \alpha , \\ \displaystyle -i\vert \nabla\vert^\alpha(k) \sum_{\ell \leq r -q} \frac{g^{(\ell)}(0)}{\ell !} \, \Psi^{r}_{n,q}\left( {\cal L}_{\tiny{\text{dom}}} ,\ell\right)(\tau), \quad \text{otherwise}. \\ \end{array} \right. \end{align} $$

Thereby, we set for $ \left (r - q - \ell +1 \right ) \deg ({\cal L}_{\tiny {\text {dom}}}) + \ell \deg ({\cal L}_{\tiny {\text {low}}}) + \alpha> n $

(85) $$ \begin{align} \Psi^{r}_{n,q}\left( {\cal L}_{\tiny{\text{dom}}},\ell \right)(\tau) = \int_0^{\tau} \xi^{\ell+q} f(\xi) d \xi. \end{align} $$

Otherwise,

(86) $$ \begin{align} \Psi^{r}_{n,q}\left( {\cal L}_{\tiny{\text{dom}}},\ell \right)(\tau) = \sum_{m \leq r - q - \ell} \frac{f^{(m)}(0)}{m!} \int_0^{\tau} \xi^{\ell+m+q} d \xi. \end{align} $$

Here $ \deg ({\cal L}_{\tiny {\text {dom}}})$ and $\deg ({\cal L}_{\tiny {\text {low}}}) $ denote the degree of the polynomials $ {\cal L}_{\tiny {\text {dom}}} $ and $ {\cal L}_{\tiny {\text {low}}} $ , respectively, and $\vert \nabla \vert ^\alpha (k) = \prod _{\alpha = \sum \gamma _j < \deg ({\cal L})} k_j^{\gamma _j}$ (cf. (8)). If $ r <q $ , the map $ {{\cal K}}^{k,r}_{o_2} ( { G},n)(\tau )$ is equal to zero.

Example 18. We consider $P_{{\mathfrak {t}}_2}( \lambda ) = - \lambda ^2$ , $p = 0$ , $ \alpha =0 $ , $ k = -k_1+k_2+k_3 $ and

$$ \begin{align*} { G}(\xi) = \xi e^{i \xi ( k_1^2 - k_2^2 - k_3^2 )}. \end{align*} $$

With the notation of Definition 3.1, we observe that $q = 1$ and $P(k_1,k_2,k_3) = k_1^2 - k_2^2 - k_3^2$ such that

$$ \begin{align*} P_{o_2}(k) &+ P = (-1)^{p} P_{{\mathfrak{t}}_2}( (-1)^{p} k) + P\\ & = (-k_1+k_2+k_3)^2 + k_1^2 - k_2^2 - k_3^2 = 2k_1^2 - 2 k_1 (k_2+k_3) + 2 k_2 k_3. \end{align*} $$

Hence,

$$ \begin{align*} {\cal L}_{\tiny{\text{dom}}} = 2 k_1^2, \quad {\cal L}_{\tiny{\text{low }}} = - 2 k_1 (k_2+k_3) + 2 k_2 k_3; \end{align*} $$

cf. also the Schrödinger Example 8. Furthermore, we observe as $\deg ({\cal L}_{\tiny {\text {dom}}}) = 2$ , $\deg ({\cal L}_{\tiny {\text {low}}}) = 1$ and $q = 1$ that

(89) $$ \begin{align} \left(r - q - \ell+1 \right) \deg({\cal L}_{\tiny{\text{dom}}}) + \ell \deg({\cal L}_{\tiny{\text{low}}})> n \quad \text{ if }\quad 2r - n > \ell. \end{align} $$

In the following we will also exploit that $f(0) = g(0) = 1$ .

• • Case ${r = 0:}$ As $ r< q = 1$ we have for all n that

  $$ \begin{align*} {{\cal K}}^{k,0}_{o_2} ( { G},n)(\tau) = 0. \end{align*} $$

• • Case $r = 1$ : For $n = 1$ we obtain

  $$ \begin{align*} {{\cal K}}^{k,1}_{o_2} ( { G},n)(\tau) = -i \Psi^{1}_{n,1}\left( {\cal L}_{\tiny{\text{dom}}},0 \right) = \int_0^\tau \xi f(\xi) d\xi = \frac{1}{2i k_1^2}\left( \tau e^ {2i \tau k_1^2}- \frac{ e^{2 i \tau k_1^2}-1}{2 i k_1^2}\right) \end{align*} $$
  as condition (89) takes for $\ell = 0$ the form $2-n> 0$ .

  On the other hand, for $n \geq 2$ we have that $n \geq \deg ({\cal L}_{\tiny {\text {dom}}})$ such that

  $$ \begin{align*} {{\cal K}}^{k,1}_{o_2} ( { G},n)(\tau) = -i { \tilde g}(0) \int_0^\tau \xi d\xi =- i \frac{\tau^2}{2}. \end{align*} $$

• • Case $r = 2$ : If $ n \geq 4$ we obtain

  $$ \begin{align*} {{\cal K}}^{k,2}_{o_2} ( { G},n)(\tau) = - i \left( \frac{\tau^2}{2} + \frac{ { \tilde g}(\tau) - 1}{\tau} \frac{\tau^3}{2}\right). \end{align*} $$

  Let $n \leq 3$ . We have that

  $$ \begin{align*} {{\cal K}}^{k,2}_{o_2} ( { G},n)(\tau) & = -i \left( \Psi^{2}_{n,1}\left( {\cal L}_{\tiny{\text{dom}}},0 \right) + \frac{g(\tau)-1}{\tau} \Psi^{2}_{n,1}\left( {\cal L}_{\tiny{\text{dom}}},1 \right) \right) \\ & = -i \left(\Psi^{2}_{n,1}\left( {\cal L}_{\tiny{\text{dom}}},0 \right) + \frac{g(\tau)-1}{\tau} \Psi^{2}_{n,1}\left( {\cal L}_{\tiny{\text{dom}}},1 \right)\right) \end{align*} $$
  and condition (89) takes the form $4-n> \ell $ .

  If $\ell = 1$ , we thus obtain for $n =1,2$

  $$ \begin{align*} \Psi^{2}_{n \leq 2 ,1}\left( {\cal L}_{\tiny{\text{dom}}},1 \right) = \int_0^\tau \xi^2 f(\xi) d\xi = \frac{ \tau^2 }{2 i k_2^2} \left(e^{2i \tau k_1^2} - 2 \Psi^{1}_{1,1}\left( {\cal L}_{\tiny{\text{dom}}},0 \right) \right) \end{align*} $$
  and for $n = 3$
  $$ \begin{align*} \Psi^{2}_{n>2,1}\left( {\cal L}_{\tiny{\text{dom}}},1 \right) = f(0)\int_0^\tau \xi^{2}d\xi = \frac{\tau^3}{3}. \end{align*} $$

  If $\ell = 0$ , on the other hand, condition (89) holds for $n = 1,2,3$ . Henceforth, we have that

  $$ \begin{align*} \Psi^{2}_{n \leq 3,1}\left( {\cal L}_{\tiny{\text{dom}}},0 \right) = \int_0^\tau \xi f(\xi)d\xi = \Psi^{1}_{1,1}\left( {\cal L}_{\tiny{\text{dom}}},0 \right). \end{align*} $$

Lemma 3.3. We keep the notations of Definition 3.1. We suppose that $ q \leq r$ ; then one has

(90) $$ \begin{align} - i \vert \nabla\vert^{\alpha} (k) \int_{0}^{\tau} \xi^{q} { e^{i \xi \left ({\cal L}_{\tiny{\text{dom}}} + {\cal L}_{\tiny{\text{low}}}\right)}} d\xi -{{\cal K}}^{k,r}_{o_2} ( { G},n)(\tau) = {{\cal O}}(\tau^{r+2} k^{\bar n}) \end{align} $$

where $ \bar n = \max (n, \deg ({\cal L}_{\tiny {\text {low}}}^{r-q +1}) + \alpha ) $ .

Proof

Recall the notation of Definition 3.1, which implies that

$$ \begin{align*}\int_{0}^{\tau} \xi^{q} { e^{i \xi \left ({\cal L}_{\tiny{\text{dom}}} + {\cal L}_{\tiny{\text{low}}}\right)}} d\xi = \int_{0}^{\tau} \xi^{q} f(\xi) g(\xi) d\xi \end{align*} $$

with $f(\xi ) = e^{i \xi {\cal L}_{\tiny {\text {dom}}}}$ and $ g(\xi ) = e^{i \xi {\cal L}_{\tiny {\text {low}}}}$ . It is just a consequence of Taylor expanding the functions $ g, { \tilde g}$ and $ f $ . If $ n \geq \text {deg}\left (\mathcal {L}_{\tiny {\text {dom}}}^{r}\right ) + \alpha $ , we have

$$ \begin{align*} -i \vert \nabla\vert^{\alpha} (k) & \int_{0}^{\tau} \xi^{q} f(\xi) g(\xi) d\xi +i \vert \nabla\vert^{\alpha} (k) \sum_{\ell \leq r - q} \frac{ { \tilde g}^{(\ell)}(0)}{\ell!} \int_0^{\tau} \xi^{\ell+q} d \xi \\ & = {{\cal O}}( \tau^{r+2} \vert \nabla\vert^{\alpha} (k) \mathcal{L}_{\tiny{\text{dom}}}^{r+1})\\ & = {{\cal O}}(\tau^{r+2} k^{\bar n}). \end{align*} $$

Else, we get

$$ \begin{align*} - i \vert \nabla\vert^{\alpha} (k) & \int_{0}^{\tau} \xi^{q} f(\xi) g(\xi) d\xi + i \vert \nabla\vert^{\alpha} (k) \sum_{\ell \leq r - q} \frac{g^{(\ell)}(0)}{\ell!} \int_0^{\tau} \xi^{\ell+q} f(\xi) d \xi \\ & = {{\cal O}}(\tau^{r+2} \vert \nabla\vert^{\alpha} (k) g^{(r-q+1)}) \\ & = {{\cal O}}(\tau^{r+2} \vert \nabla\vert^{\alpha} (k) {\cal L}_{\tiny{\text{low}}}^{r-q+1}) \end{align*} $$

where the latter follows from the observation that $ g^{(\ell )}(\xi ) = (i {\cal L}_{\tiny {\text {low}}})^{\ell } e^{i \xi {\cal L}_{\tiny {\text {low}}}} $ .

If, on the other hand, $ \left (r- q - \ell +1 \right ) \deg ({\cal L}_{\tiny {\text {dom}}}) + \ell \deg ({\cal L}_{\tiny {\text {low}}}) + \alpha \leq n $ , then

$$ \begin{align*} & - i \vert \nabla\vert^{\alpha} (k) \frac{g^{(\ell)}(0)}{\ell!} \int_0^{\tau} \xi^{\ell+q} f(\xi) d \xi +i \vert \nabla\vert^{\alpha} (k)\frac{g^{(\ell)}(0)}{\ell!} \sum_{m \leq r - q - \ell} \frac{f^{(m)}(0)}{m!} \int_0^{\tau} \xi^{\ell+m+q} d \xi \\ & = \frac{g^{(\ell)}(0)}{\ell!} \int_0^{\tau} \xi^{\ell+q} {{\cal O}} \left(\xi^{r-q-\ell+1} \vert \nabla\vert^{\alpha} (k){\cal L}_{\tiny{\text{dom}}}^{r-q-\ell+1} \right) d \xi \\ & = {{\cal O}}(\tau^{r+2} \vert \nabla\vert^{\alpha} (k) {\cal L}_{\tiny{\text{low}}}^{\ell} {\cal L}_{\tiny{\text{dom}}}^{r-q-\ell+1} )\\ & = {{\cal O}}(\tau^{r+2} k^n ), \end{align*} $$

which allows us to conclude.

Remark 3.4. In the proof of Lemma 3.3, one has

$$ \begin{align*} \deg \left( {\cal L}_{\tiny{\text{low}}}^{\ell} {\cal L}_{\tiny{\text{dom}}}^{r-q-\ell+1} \right) \geq\deg \left( {\cal L}_{\tiny{\text{low}}}^{r-q+1} \right). \end{align*} $$

If $ n = \deg \left ( {\cal L}_{\tiny {\text {low}}}^{r-q+1} \right ) + \alpha $ , we cannot carry out a Taylor series expansion of $ f $ and we have to perform the integration exactly. This will give a more complicate numerical scheme. If, on the other hand, n is larger, part of the Taylor expansions of $ f $ will be possible. In fact, n corresponds to the regularity of the solution we assume a priori; see Remark 1.1.

3.2 A Birkhoff type factorisation

The character $ \Pi ^n : {\mathcal {H}} \rightarrow {{\cal C}} $ is quite complex since one needs to compute several nonlinear interactions (oscillations) at the same time. Indeed, most of the time the operator $ {{\cal K}}^{k,r}_{o_2} \left ( \cdot ,n \right ) $ is applied to a linear combination of monomials of the form $ e^{i \xi P_j(k)} $ . We want to single out every oscillation through a factorisation of this character. We start by introducing a splitting with a projection $ {\mathcal {Q}} $ :

$$ \begin{align*} {{\cal C}} = {{\cal C}}_- \oplus {{\cal C}}_+, \quad {\mathcal{Q}} : {{\cal C}} \rightarrow {{\cal C}}_- \end{align*} $$

where $ {{\cal C}}_- $ is the space of polynomials $ Q(\xi ) $ and $ {{\cal C}}_+ $ is the subspace of functions of the form $ z \mapsto \sum _j Q_j(z)e^{i z P_j(k_1,\ldots ,k_n) } $ with $ P_j \neq 0 $ .

Remark 3.6. In the classical Birkhoff factorisation for Laurent series, we consider $A = \mathbf {C}[[t,t^{-1}]$ , with finite pole part. In this context, the splitting reads $ A = A_- \oplus A_+ $ where $A_- = t^{-1}\mathbf {C}[t^{-1}]$ and $A_+ = \mathbf {C}[[t]]$ , such that $ {\mathcal {Q}} $ keeps only the pole part of a series:

$$ \begin{align*} {\mathcal{Q}}\Big( \sum_{n} a_n t^{n} \Big) = \sum_{ n< 0} a_n t^{n} \in A_-. \end{align*} $$

The idea is to remove the divergent part of the series; that is, its pole part. In our context, the structure of the factorisation is quite different, as we are interested in singling out the oscillations. Let us suppose that we start with a term of the form $ z \mapsto e^{i z P(k_1,\ldots ,k_n) }$ , where P corresponds to the dominant part of some differential operator. Then, the integral over this term yields two contributions:

$$ \begin{align*} \int_{0}^{t} e^{i \xi P(k_1,\ldots ,k_n)} d\xi = \frac{ e^{i t P(k_1,\ldots ,k_n)} - 1}{i \small{P(k_1,\ldots ,k_n)}}. \end{align*} $$

One is the oscillation $e^{i z P(k_1,\ldots ,k_n) }$ we start with evaluated at time $z = t$ and the other one is the constant term $ -1 $ . These terms are obtained by applying recursively the projection $ {\mathcal {Q}} $ . This approach seems new from the literature, and it is quite different in spirit from what has been observed for singular SPDEs.

We set

(91) $$ \begin{align} {{\cal K}}^{k,r}_{o_2,-}:= {\mathcal{Q}} \circ {{\cal K}}^{k,r}_{o_2} , \quad {{\cal K}}^{k,r}_{o_2,+}:=\left( {\mathrm{id}} - {\mathcal{Q}} \right) \circ {{\cal K}}^{k,r}_{o_2}. \end{align} $$

One has

$$ \begin{align*} {{\cal K}}^{k,r}_{o_2} = {{\cal K}}^{k,r}_{o_2,-} + {{\cal K}}^{k,r}_{o_2,+}. \end{align*} $$

We define a character $ A^n : {\mathcal {H}}_+ \rightarrow \mathbf {C} $ by

(92) $$ \begin{align} \begin{split} A^n( {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F)) & = \left( {\mathcal{Q}} \circ \partial^{m} \Pi^{n} {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} F) \right)(0), \end{split} \end{align} $$

where $\partial ^{m} \Pi ^{n} {\cal I}^{r}_{o_2}( \lambda _{k}^{\ell } F)$ is the $ m $ th derivative of the function $ t \mapsto (\Pi ^{n} {\cal I}^{r}_{o_2}( \lambda _{k}^{\ell } F))(t) $ . The character $ A^n $ applied to $ {\cal I}^{(r,m)}_{o_2}( \lambda _{k}^{\ell } F) $ is extracting the coefficient of $ \tau ^{m} $ multiplied by $ m! $ in $ \Pi ^n {\cal I}^{r}_{o_2}( \lambda _{k}^{\ell } F) $ . If we extend $ \Pi ^n $ to $ {\mathcal {H}}_+ $ by setting

$$ \begin{align*} \Pi^n( {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F)) = \partial^{m} \Pi^{n} {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} F), \end{align*} $$

then we have a new expression for $ A^n $ ,

(93) $$ \begin{align} A^n = \left( {\mathcal{Q}} \circ \Pi^n \cdot \right)(0). \end{align} $$

We define a character $ \hat \Pi ^n : {\mathcal {H}} \rightarrow {{\cal C}} $ which computes only one interaction by repeatedly applying the projection $ {\mathrm {id}} - {\mathcal {Q}} $

(94) $$ \begin{align} \begin{split} \hat{\Pi}^n \left( F \cdot \bar F \right)(\tau) & = \left( \hat{\Pi}^n F \right)(\tau) \left( \hat{\Pi}^n \bar F \right)(\tau), \quad (\hat{\Pi}^n \lambda^{\ell})(\tau) = \tau^{\ell}, \\ \hat{\Pi}^n({\cal I}^{r}_{o_1}( \lambda_{k}^{\ell} F ))(\tau) & = \tau^{\ell }e^{i \tau P_{o_1}(k)} \hat{\Pi}^n ({\cal D}^{r-\ell}(F))(\tau), \\ \hat \Pi^n({\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} F )) & = {{\cal K}}_{o_2,+}^{k,r}( \hat \Pi^n( \lambda^{\ell} {\cal D}^{r-\ell-1}(F)),n ) \end{split} \end{align} $$

for $ F, \bar {F} \in {\mathcal {H}} $ . One can notice that $ \hat \Pi ^n $ takes value in $ {{\cal C}}_+ $ on elements of the form ${\cal I}^{r}_{o_2}( \lambda _{k}^{\ell } F ) $ . The character $ \hat {\Pi }^n $ is central for deriving the local error analysis for the numerical scheme. It has nice properties outlined in Proposition 3.9 which are crucial for proving Theorem 3.17. We provide an identity for the approximation $ \Pi ^n $ :

(95) $$ \begin{align} \begin{split} \Pi^n = \left( \hat \Pi^n \otimes A^n \right) \Delta. \end{split} \end{align} $$

In the identity above, we do not need a multiplication because $ A^n(F) \in \mathbf {C} $ for every $ F \in {\mathcal {H}}_+ $ and $ {{\cal C}}$ is a $ \mathbf {C} $ -vector space. Therefore, we use the identification $ \mathcal {C} \otimes \mathbf {C} \cong \mathcal {C} $ .

Proof

We prove this identity by induction on the number of edges of a forest. We first consider a tree of the form $ {\cal I}^{r}_{o_1}( \lambda _k^{\ell } F) $ ; then we get

$$ \begin{align*} \begin{split} \left( \hat \Pi^n(\cdot)(\tau) \otimes A^n \right) \Delta {\cal I}^{r}_{o_1}( \lambda_k^{\ell} F ) & = \left(\hat \Pi^n \left( {\cal I}^{r}_{o_1}( \lambda_k^{\ell} \cdot ) \right)(\tau) \otimes A^n \right) \Delta {\cal D}^{r - \ell}(F) \\ & = \tau^{\ell} e^{i \tau P_{o_1}(k)} \left( \hat \Pi^n \left( \cdot \right)(\tau) \otimes A^n \right) \Delta {\cal D}^{r - \ell}(F) \\ & = \tau^{\ell} e^{i \tau P_{o_1}(k)} (\Pi^n {\cal D}^{r - \ell}(F))(\tau) \\ & = \left( \Pi^n {\cal I}^{r}_{o_1}( \lambda_k^{\ell} F ) \right)(\tau), \end{split} \end{align*} $$

where we have used our inductive hypothesis. We look now at a tree of the form $ {\cal I}^{r}_{o_2}( \lambda _{k}^{\ell } F) $ :

$$ \begin{align*} \begin{split} & \left( \hat \Pi^n \otimes A^n \right) \Delta {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} F) = \left( \hat \Pi^n({\cal I}^{r}_{o_2}( \lambda_k^{\ell} \cdot)) \otimes A^n \right) \Delta {\cal D}^{r-\ell-1}(F) \\& + \sum_{m \leq r + 1} \frac{1}{m!}\hat{\Pi}^n( \lambda^{m}) A^n( {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) ) \\ & = {{\cal K}}_{o_2,+}^{k,r} \left( \hat \Pi^n( \lambda^{\ell} \cdot) \otimes A^n, n \right) \Delta {\cal D}^{r-\ell-1}(F) + {{\cal K}}_{o_2,-}^{k,r}(\Pi^n \lambda^{\ell} {\cal D}^{r -\ell-1}(F),n) \\ & = {{\cal K}}_{o_2,+}^{k,r}(\Pi^n \lambda^{\ell} {\cal D}^{r-\ell-1}(F),n) + {{\cal K}}_{o_2,-}^{k,r}(\Pi^n \lambda^{\ell} {\cal D}^{r-\ell-1}(F),n) \\ & = \Pi^n \left({\cal I}^{r}_{o_2}( \lambda^{\ell}_{k} F) \right) \end{split} \end{align*} $$

where we used the following identification

$$ \begin{align*} (\hat \Pi^n \otimes A^n) (F_1 \otimes F_2) = ( \hat \Pi^n F_1 \otimes A^n F_2 ) = \hat \Pi^n(F_1) A^n(F_2) \end{align*} $$

and that

$$ \begin{align*} & \sum_{m \leq r+1} \frac{1}{m!}\hat{\Pi}^n( \lambda^{m}) \, A^n( {\cal I}^{(r,m)}_{o_2}( \lambda_{k}^{\ell} F) ) \\ & = \sum_{m \leq r + 1} \frac{1}{m!}\hat{\Pi}^n( \lambda^{m}) \,\left( {\mathcal{Q}} \circ \partial^{m} \Pi^n {\cal I}^{r}_{o_2}( \lambda_{k}^{\ell} F) \right)(0) \\ & = {{\cal K}}_{o_2,-}^{k,r}(\Pi^n \lambda^{\ell} {\cal D}^{r - 1-\ell}(F),n). \end{align*} $$

The interest of the decomposition given by Proposition 3.7 comes from Proposition 3.9: $\hat {\Pi }^n$ only involves one oscillation.

Example 19. We compute $ A^n $ and $ \hat \Pi ^n $ for the following decorated trees that appear for the cubic NLS equation (see also (59)):

where $ \ell = -k_1 + k_2 + k_3 $ . Then, when $ r = 0 $ and $ n < 2 $ , one has from (132)

$$ \begin{align*} (\Pi^n {\cal D}^r(T_1))(\tau) = -i \frac{e^{2 i \tau k_1^2}-1}{2 i k_1^2}. \end{align*} $$

On the other hand,

$$ \begin{align*} \Delta {\cal D}^{0}(T_1) = {\cal D}^{0}(T_1) \otimes {\mathbf{1}} + {\mathbf{1}} \otimes \hat {\cal D}^{(0,0)}(T_1) \end{align*} $$

and

$$ \begin{align*} \hat \Pi^{n}({\cal D}^{0}(T_1)) = -\frac{e^{2 i \tau k_1^2}}{2 k_1^2}, \quad A^n(\hat {\cal D}^{(0,0)}(T_1)) = \frac{1}{2 k_1^2}. \end{align*} $$

When $ n=2 $ , one gets

$$ \begin{align*} (\Pi^n {\cal D}^0(T_1))(\tau) = \tau, \quad \hat \Pi^{n}({\cal D}^{0}(T_1)) = 0, \quad A^n(\hat {\cal D}^{(0,0)}(T_1)) = 1. \end{align*} $$

Now, we consider the tree $ T_3 $ and we assume that $ r=1 $ and $ n =2 $ . We calculate that (for details, see (143) in Section 5)

$$ \begin{align*} (\Pi^n {\cal D}^1(T_3))(\tau) = - \frac{\tau^2}{2}. \end{align*} $$

Then,

When one applies $ (\hat \Pi ^n \otimes A^n) $ , the only nonzero contribution is given by the third term from the previous computation:

$$ \begin{align*} (\hat \Pi^n \lambda^2)(\tau) = \tau^2, \quad A^n ( \hat {\cal D}^{(1,2)}(T_3)) = -1. \end{align*} $$

In order to see a nontrivial interaction in the previous term, one has to consider a higher-order approximation such as for $ r = 2 $ and $ n=3 $ . In this case, one has

(96) $$ \begin{align} (\Pi^{n} {\cal D}^{1}(T_1) )(\tau) & = - i \int_{0}^{\tau} e^{2i s k_1^2} ds + \frac{\tau^2}{2} \mathscr{F}_{\tiny{\text{low}}} \left( T_1 \right) \\& = - \frac{e^{2i \tau k_1^2} }{2 k_1^2} + \frac{1}{2 k_1^2} + \frac{\tau^2}{2} \mathscr{F}_{\tiny{\text{low}}} \left( T_1 \right). \notag \end{align} $$

Then,

$$ \begin{align*} A^n(\hat {\cal D}^{(1,0)}(T_1)) = \frac{1}{2k_1^2}, \quad A^n(\hat {\cal D}^{(1,1)}(T_1)) = 0, \quad A^n(\hat {\cal D}^{(1,2)}(T_1)) = \mathscr{F}_{\tiny{\text{low}}} \left( T_1 \right). \end{align*} $$

Next, one has to approximate

$$ \begin{align*} - i \int_0^\tau \mathrm{e}^{i s k^2} \Big( \mathrm{e}^{i s( k_4^2 - k_5^2 - k_{123}^2)} (\Pi^n {\cal D}^1(T_1))(s) \Big) d s \end{align*} $$

where $ k = -k_1 + k_2 + k_3 - k_4 + k_5 $ and $ k_{123} = -k_1 + k_2 + k_3 $ . Thanks to the structure of $(\Pi ^n {\cal D}^1(T_1))(s)$ (see (96)), it remains to control the following two oscillations:

$$ \begin{align*} k^2 + k_4^2 - k_5^2 - k_{123}^2 & =\mathscr{F}_{\tiny{\text{dom}}}\left( T_2 \right) + \mathscr{F}_{\tiny{\text{low}}} \left( T_2 \right) \\ k^2 + k_4^2 - k_5^2 - k_{123}^2 + 2k_1^2 & = \mathscr{F}_{\tiny{\text{dom}}}\left( T_3 \right) + \mathscr{F}_{\tiny{\text{low}}} \left( T_3 \right). \end{align*} $$

Then

$$ \begin{align*} (\Pi^{n} {\cal D}^2(T_3))(\tau) & = - \int_{0}^{\tau} \frac{e^{i s \mathscr{F}_{\tiny{\text{dom}}}\left( T_3 \right) }}{2ik_1^2} \left( 1+ i \mathscr{F}_{\tiny{\text{low}}} \left( T_3 \right) s - \mathscr{F}_{\tiny{\text{low}}} \left( T_3 \right)^2 \frac{s^2}{2}\right) ds \\ &\quad{} + \int_{0}^{\tau} \frac{e^{i s\mathscr{F}_{\tiny{\text{dom}}}\left( T_2 \right)}}{2ik_1^2} \left( 1+ i \mathscr{F}_{\tiny{\text{low}}} \left( T_2 \right) s - \mathscr{F}_{\tiny{\text{low}}} \left( T_2 \right)^2 \frac{s^2}{2}\right) ds - i \frac{\tau^3}{3 !} \mathscr{F}_{\tiny{\text{low}}} \left( T_1 \right). \end{align*} $$

One has the following identities:

In the next proposition, we write a Birkhoff type factorisation for the character $ \hat \Pi ^n $ defined from $ \Pi ^n $ and the antipode. Such an identity was also obtained in the context of SPDEs (see [Reference Bruned, Hairer and Zambotti13]) but with a twisted antipode. Our formulation is slightly simpler due to the simplifications observed at the level of the algebra (see Remark 2.11). The Proposition 3.8 is not used in the sequel but it gives an inductive way to compute $ \hat \Pi ^n $ in terms of $ \Pi ^n $ . It can be seen as a rewriting of identity (95).

Proposition 3.8. One has

(97) $$ \begin{align} \hat \Pi^n = \left( \Pi^n \otimes ({\mathcal{Q}} \circ \Pi^n {\mathcal{A}} \cdot)(0) \right) \Delta. \end{align} $$

Proof

From (95), one gets

(98) $$ \begin{align} \Pi^n = \left( \hat \Pi^n \otimes ({\mathcal{Q}} \circ \Pi^n \cdot)(0) \right) \Delta = \hat \Pi^n * ({\mathcal{Q}} \circ \Pi^n \cdot)(0) \end{align} $$

where the product $ * $ is defined from the coaction $ \Delta $ . Then, if we multiply the identity (98) by the inverse $ ({\mathcal {Q}} \circ \Pi ^n {\mathcal {A}} \cdot )(0) $ , we get

$$ \begin{align*} \Pi^n * ({\mathcal{Q}} \circ \Pi^n {\mathcal{A}} \cdot)(0) & =\left( \hat \Pi^n * ({\mathcal{Q}} \circ \Pi^n \cdot)(0) \right) * ({\mathcal{Q}} \circ \Pi^n {\mathcal{A}} \cdot)(0) \\ & = \left( \left( \hat \Pi^n \otimes ({\mathcal{Q}} \circ \Pi^n \cdot)(0) \right) \Delta \otimes ({\mathcal{Q}} \circ \Pi^n {\mathcal{A}} \cdot)(0) \right) \Delta \\ & = \left( \hat \Pi^n \otimes \left( ({\mathcal{Q}} \circ \Pi^n \cdot)(0) \otimes ({\mathcal{Q}} \circ \Pi^n {\mathcal{A}} \cdot)(0) \right) {\Delta^{\!+}} \right) \Delta \\ & = \hat \Pi^n \end{align*} $$

where we have used

$$ \begin{align*} \left( \Delta \otimes {\mathrm{id}} \right) \Delta = \left( {\mathrm{id}} \otimes {\Delta^{\!+}} \right) \Delta, \quad {\mathcal{M}} \left( {\mathrm{id}} \otimes {\mathcal{A}} \right) {\Delta^{\!+}} = {\mathbf{1}} {\mathbf{1}}^{\star}. \end{align*} $$

This concludes the proof.

3.3 Local error analysis

In this section, we explore the properties of the character $ \hat \Pi ^n $ which allow us to conduct the local error analysis of the approximation given by $ \Pi ^n $ . Proposition 3.9 shows that only one oscillation is treated through $ \hat \Pi ^n $ .

Proposition 3.9. For every forest $ F \in \hat {\mathcal {H}} $ , there exists a polynomial $ B^n\left ( {\cal D}^r(F) \right ){ (\xi )} $ such that

(99) $$ \begin{align} \hat \Pi^n\left( {\cal D}^r(F) \right)(\xi) = B^n\left( {\cal D}^r(F) \right)(\xi) e^{i \xi\mathscr{F}_{\tiny{\text{dom}}}( F)} \end{align} $$

where $ \mathscr {F}_{\tiny {\text {dom}}}(F) $ is given in Definition 2.6. Moreover, $ B^n({\cal D}^r(F))(\xi ) $ is given by

(100) $$ \begin{align} B^n({\cal D}^r(F))(\xi) = \frac{P(\xi)}{Q}, \quad Q = \prod_{\bar T \in A} \left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right)^{m_{\bar T}} \end{align} $$

where $ P(\xi ) $ is a polynomial in $ \xi $ and the $k_i$ , $ A $ is a set of decorated subtrees of $ F $ satisfying the same property as in Corollary 2.9.

Proof

We proceed by induction. We get

$$ \begin{align*} \hat{\Pi}^n\left( {\cal D}^r(F \cdot \bar F) \right)(\xi) & = \hat{\Pi}^n\left( {\cal D}^r(F ) \right)(\xi) \hat{\Pi}^n\left( {\cal D}^r( \bar F) \right)(\xi) \\ & = B^n\left( {\cal D}^r(F) \right)(\xi) e^{i \xi\mathscr{F}_{\tiny{\text{dom}}}( F)} B^n\left( {\cal D}^r(\bar F) \right)(\xi) e^{i \xi\mathscr{F}_{\tiny{\text{dom}}}( \bar F)} \\ & = B^n\left( {\cal D}^r(F) \right)(\xi) B^n\left( {\cal D}^r(\bar F) \right)(\xi) e^{i \xi(\mathscr{F}_{\tiny{\text{dom}}}( F) +\mathscr{F}_{\tiny{\text{dom}}}( \bar F) )} \\ & = B^n\left( {\cal D}^r(F \cdot \bar F) \right)(\xi) e^{i \xi\mathscr{F}_{\tiny{\text{dom}}}( F \cdot \bar F)}. \end{align*} $$

The pointwise product preserves the structure given by (100).

Then for $ T = {\cal I}_{o_1}( \lambda _{k}^{\ell }F) $ , one gets by the definition of $\hat {\Pi }^n$ given in (94) that

$$ \begin{align*} \hat{\Pi}^n\left( {\cal D}^r(T) \right)(\xi) & = \xi^{\ell} e^{i \xi P_{o_1}(k) } \hat{\Pi}^n\left( {\cal D}^{r-\ell}(F) \right)(\xi) \\ & = \xi^{\ell} e^{i \xi P_{o_1}(k) +i \xi\mathscr{F}_{\tiny{\text{dom}}}( F) } B^n\left( {\cal D}^r( F) \right)(\xi) \\ & = B^n\left( {\cal D}^r(T) \right)(\xi) e^{i \xi\mathscr{F}_{\tiny{\text{dom}}}( T)}. \end{align*} $$

One gets $ B^n\left ( {\cal D}^r(T) \right )(\xi ) = \xi ^{\ell } B^n\left ( {\cal D}^r(F) \right )(\xi ) $ and can conclude on the preservation of the factorisation (100). We end with the decorated tree $ T = {\cal I}_{o_2}( \lambda _{k}^{\ell } F) $ . By the definition of $\hat {\Pi }^n$ given in (94), we obtain that

$$ \begin{align*} \hat \Pi^n\left( T \right)(\xi) = {{\cal K}}_{o_2,+}^{k,r}( \hat \Pi^n( \lambda^{\ell} {\cal D}^{r-\ell-1}(F)),n )(\xi). \end{align*} $$

Now, we apply the induction hypothesis on $ F $ , which yields

$$ \begin{align*} \hat \Pi^n( {\cal D}^{r-\ell-1}(F))(\xi) = B^n( {\cal D}^{r-\ell-1}( F))(\xi) e^{i \xi\mathscr{F}_{\tiny{\text{dom}}}(F)} \end{align*} $$

and we conclude by applying Definition 3.1. Indeed, by applying $ {{\cal K}}_{o_2,+}^{k,r} $ to $ \xi ^{\ell } e^{i \xi \mathscr {F}_{\tiny {\text {dom}}}( F)} $ , we can get in (85) extras terms $ \frac {1}{Q} $ coming from expressions of the form

$$ \begin{align*} \int_0^{\tau} \xi^{\ell+q} e^{i \xi\mathscr{F}_{\tiny{\text{dom}}}(T)} d \xi. \end{align*} $$

Then, by computing this integral, we obtain coefficients of the form

$$ \begin{align*} \frac{1}{(i\mathscr{F}_{\tiny{\text{dom}}}(T))^{m_T}} \end{align*} $$

if $\mathscr {F}_{\tiny {\text {dom}}}(T)^{m_T} \neq 0 $ . This term will be multiplied by $ B^n( {\cal D}^{r-\ell -1}( F))(\xi ) e^{i \xi \mathscr {F}_{\tiny {\text {dom}}}(F)} $ and preserves the structure given in (100). When performing the Taylor series expansion by applying $ {{\cal K}}_{o_2,+}^{k,r} $ (cf. (84) and (86)), we can also get some extra polynomials in the $ k_i $ . This leads to the factor $P(\xi )$ in (100) and concludes the proof.

The next recursive definition introduces a systematic way to compute the local error from the structure of the decorated tree and the coaction $ \Delta $ .

Definition 3.11. Let $ n \in \mathbf {N} $ , $ r \in {\mathbf {Z}} $ . We recursively define $ \mathcal {L}^{r}_{\tiny {\text {low}}}(\cdot ,n)$ as

$$ \begin{align*} \mathcal{L}^{r}_{\tiny{\text{low}}}(F,n) = 1, \quad r < 0. \end{align*} $$

Else,

$$ \begin{align*} &\mathcal{L}^{r}_{\tiny{\text{low}}}({\mathbf{1}},n) = 1, \quad \mathcal{L}^{r}_{\tiny{\text{low}}}(F \cdot \bar F,n) = \mathcal{L}^{r}_{\tiny{\text{low}}}(F,n ) + \mathcal{L}^{r}_{\tiny{\text{low}}}( \bar F,n) \\ &\qquad \qquad \qquad \quad\quad \mathcal{L}^{r}_{\tiny{\text{low}}}({\cal I}_{o_1}( \lambda_{k}^{\ell} F ),n) = \mathcal{L}^{r-\ell}_{\tiny{\text{low}}}( F,n ) \\ &\mathcal{L}^{r}_{\tiny{\text{low}}}({\cal I}_{o_2}( \lambda^{\ell}_{k} F ),n) = k^{\alpha} \mathcal{L}^{r-\ell-1}_{\tiny{\text{low}}}( F,n ) + {\mathbf{1}}_{\lbrace r-\ell \geq 0 \rbrace} \sum_j k^{\bar n_j} \end{align*} $$

where

$$ \begin{align*} \bar n_j = \max_{ m}\left(n,\deg\left( P_{(F^{(1)}_j, F^{(2)}_j,m)} \mathscr{F}_{\tiny{\text{low}}} ({\cal I}_{({\mathfrak{t}}_2,p)}( \lambda^{\ell}_{k} F^{(1)}_j ))^{r-\ell +1- m} + \alpha \right) \right) \end{align*} $$

with

$$ \begin{align*} \Delta {\cal D}^{r-\ell-1}(F) & = \sum_{j} F^{(1)}_j \otimes F^{(2)}_j, \\ \quad A^n(F^{(2)}_j) B^n\left( F^{(1)}_j \right)(\xi) & = \sum_{m \leq r-\ell -1} \frac{P_{(F^{(1)}_j, F^{(2)}_j,m)}}{Q_{(F^{(1)}_j, F^{(2)}_j,m)}}\xi^m \end{align*} $$

and $ \mathscr {F}_{\tiny {\text {low}}} $ is defined in Definition 2.6.

Remark 3.14. In Section 5 we derive the low regularity resonance-based schemes on concrete examples up to order 2. In the discussed examples, one does not get any contribution from $ B^n $ and $ A^n $ (see also Example 19). Thus, one can work with a simplify definition of $\bar n_j $ given by

$$ \begin{align*} \bar n_j = \max(n, \deg( \mathscr{F}_{\tiny{\text{low}}} ({\cal I}_{({\mathfrak{t}}_2,p)}( \lambda^{\ell}_{k} F_j ))^{r-\ell +1} + \alpha ), \end{align*} $$

where

$$ \begin{align*} \sum_j F_j = {\mathcal{M}}_{(1)} \Delta {\cal D}^{r-\ell-1}(F), \quad F_j \in H , \quad {\mathcal{M}}_{(1)} \left( F_1 \otimes F_2 \right) = F_1. \end{align*} $$

Now we are in the position to state the approximation error of $\Pi ^{n,r}$ to $\Pi $ (cf. (34)).

Theorem 3.17. For every $ T \in {\mathcal {T}} $ , one has

$$ \begin{align*} \left(\Pi T - \Pi^{n,r} T \right)(\tau) = \mathcal{O}\left( \tau^{r+2} \mathcal{L}^{r}_{\tiny{\text{low}}}(T,n) \right) \end{align*} $$

where $\Pi $ is defined in (82), $\Pi ^n$ is given in (83) and $\Pi ^{n,r} = \Pi ^n {\cal D}^r$ .

Proof

We proceed by induction on the size of a forest by using the recursive definition (83) of $ \Pi ^n $ and we prove a more general version of the theorem for forests. In fact, only the version for trees is needed for the local error analysis. First, one gets

$$ \begin{align*} \left(\Pi - \Pi^{n,r} \right)({\mathbf{1}})(\tau) = 0 = \mathcal{O}\left( \tau^{r+2} \mathcal{L}^{r}_{\tiny{\text{low}}}({\mathbf{1}},n) \right). \end{align*} $$

One also has

$$ \begin{align*} \left(\Pi - \Pi^{n,r} \right)({\cal I}_{o_1}( \lambda_{k}^{\ell} F ))(\tau) & = \tau^{\ell} e^{i \tau P_{o_1}(k)} (\Pi - \Pi^{n,r-\ell})(F)(\tau) \\ & = \mathcal{O}\left( \tau^{r+2} \mathcal{L}^{r-\ell}_{\tiny{\text{low}}}(F,n) \right). \end{align*} $$

Then, one gets again by (82) and (83) that

$$ \begin{align*} \left(\Pi - \Pi^{n,r} \right)(F \cdot \bar F)(\tau) & = \left(\Pi - \Pi^{n,r} \right)(F)(\tau) ( \Pi^{n,r} \bar F)(\tau) + (\Pi F)(\tau) \left(\Pi - \Pi^{n,r} \right)(\bar F)(\tau) \\ & = \mathcal{O}\left( \tau^{r+2} \mathcal{L}^{r}_{\tiny{\text{low}}}(F,n) \right) + \mathcal{O}\left( \tau^{r+2} \mathcal{L}^{r}_{\tiny{\text{low}}}(\bar F,n) \right) \\ & = {{\cal O}} \left( \tau^{r+2} \mathcal{L}^{r}_{\tiny{\text{low}}}(F \cdot \bar F,n) \right) \end{align*} $$

where we use Definition 3.11. At the end, by (82) and (83) and inserting zero in terms of

$$ \begin{align*} \pm \Pi^{n,r-\ell - 1}( F)(\xi) \end{align*} $$

as well as using that $ \Pi ^{n,r-\ell - 1}( F) = \Pi ^n {\cal D}^{r-\ell -1}(F) $ , we obtain

$$ \begin{align*} &\left( \Pi- \Pi^{n,r} \right) \left( {\cal I}_{o_2}( \lambda_k^{\ell} F) \right) (\tau) = - i \vert \nabla\vert^{\alpha} (k) \int_{0}^{\tau} \xi^{\ell} e^{i \xi P_{o_2}(k)} (\Pi - \Pi^{n,r-\ell - 1})( F)(\xi) d \xi \\ & \quad{}- i \vert \nabla\vert^{\alpha} (k)\int_{0}^{\tau} e^{i \xi P_{o_2}(k)} (\Pi^{n} { \lambda^{\ell} } {\cal D}^{r-\ell-1}(F) )(\xi) d \xi -{{\cal K}}^{k,r}_{o_2} ( \Pi^{n}( \lambda^\ell{\cal D}^{r-\ell-1} (F)),n )(\tau) \\ & = \int_{0}^{\tau} {{\cal O}} \left( \xi^{r+1} k^{\alpha} \mathcal{L}^{r-\ell-1}_{\tiny{\text{low}}}(F,n) \right) d\xi + {\mathbf{1}}_{\lbrace r-\ell \geq 1 \rbrace} \sum_{j} {{\cal O}}(\tau^{r+2} k^{\bar n_j}) \\ & = {{\cal O}} \left( \tau^{r+2} \mathcal{L}^{r}_{\tiny{\text{low}}}(F,n) \right). \end{align*} $$

Note that in the above calculation we have used the following decomposition:

$$ \begin{align*} \Pi^n \left( {\cal D}^{r-\ell-1} (F)\right) = \left( \hat \Pi^n \otimes A^n \right) \Delta {\cal D}^{r-\ell-1} (F), \end{align*} $$

which by Proposition 3.9 implies that one has

(101) $$ \begin{align} \Pi^n \left( {\cal D}^{r-\ell-1} (F)\right)(\xi) = \sum_{j} A^n(F^{(2)}_j) B^n\left( F^{(1)}_j \right)(\xi) e^{i \xi\mathscr{F}_{\tiny{\text{dom}}}( F^{(1)}_j)} \end{align} $$

where we have used Sweedler notations for the coaction $ \Delta $ . Then, one has

$$ \begin{align*} A^n(F^{(2)}_j) B^n\left( F^{(1)}_j \right)(\xi) = \sum_{m \leq r-\ell -1} \frac{P_{(F^{(1)}_j, F^{(2)}_j,m)}}{Q_{(F^{(1)}_j, F^{(2)}_j,m)}}\xi^m \end{align*} $$

where $ P_{(F^{(1)}_j, F^{(2)}_j,m)} $ and $ Q_{(F^{(1)}_j, F^{(2)}_j,m)}$ are polynomials in the frequencies $ k_1,..,k_n $ .

Thus, by applying Lemma 3.3, we get an error for every term in the sum (101) which is at most of the form

$$ \begin{align*} \sum_j {{\cal O}}(\tau^{r+2} k^{\bar n_j} ) \end{align*} $$

where

$$ \begin{align*} \bar n_j = \max_{ m}\left(n,\deg\left( P_{(F^{(1)}_j, F^{(2)}_j,m)} \mathscr{F}_{\tiny{\text{low}}} ({\cal I}_{({\mathfrak{t}}_2,p)}( \lambda^{\ell}_{k} F^{(1)}_j ))^{r-\ell +1- m} + \alpha \right) \right). \end{align*} $$

This concludes the proof.