2. Preliminaries

2.1 Orthonomic systems of PDEs

We consider a given real analytic system of PDEs on ${\mathbb{R}}^N$ in orthonomic form (see below). Independent variables $\mathbf{x} = (x^1,\ldots , x^N)$ and dependent variables $\mathbf{u}=(u^1,\ldots ,u^M)$ are used to state general results; the Einstein summation convention applies throughout. For particular examples, however, commonly used notation is adopted where this aids clarity.

Derivatives of each $u^\alpha$ are written as $u^\alpha _{\mathbf{J}}$ , where $\mathbf{J}=(j^1,\ldots ,j^N)$ is a multi-index; each $j^i$ denotes the number of derivatives with respect to $x^i$ , so $u^\alpha _{\mathbf{0}}= u^\alpha$ . The variables $x^i$ and $u^\alpha _{\mathbf{J}}$ can be regarded as coordinates on the infinite jet space (see Olver [Reference Olver16]). With this approach, the partial derivative with respect to $x^i$ is replaced by the total derivative,

\begin{equation*} D_i=\frac {{\partial }}{{\partial } x^i}+u^\alpha _{\mathbf{J} i}\,\frac {{\partial }}{{\partial } u^\alpha _{\mathbf{J}}}\,,\quad \text{where}\quad \mathbf{J} i=\left(\,j^1,\ldots ,j^{i-1},j^i+1,j^{i+1},\ldots , j^N\right). \end{equation*}

(Total derivatives commute with one another.) To keep the notation concise, let

\begin{equation*} \mathbf{D}_{\mathbf{J}}=D_1^{j^1}D_2^{j^2}\ldots D_p^{j^N},\qquad |\mathbf{J}|=j^1+\cdots +j^N. \end{equation*}

Note that $u^\alpha _{\mathbf{J}}=\mathbf{D}_{\mathbf{J}} u^\alpha$ is a $|\mathbf{J}|$ -th order derivative and $u^\alpha _{\mathbf{J}+\mathbf{K}}=\mathbf{D}_{\mathbf{K}}u^\alpha _{\mathbf{J}}$ .

From here on, we use $[\mathbf{u}]$ to represent $\mathbf{u}$ and finitely many of its total derivatives; more generally, square brackets around an expression denote the expression and as many of its total derivatives as are needed. All functions of $(\mathbf{x},[\mathbf{u}])$ are assumed to be analytic, at least locally.

The variables $u^\alpha _{\mathbf{J}}$ may be ranked using any total order $\preceq$ that satisfies the following positivity conditions:

\begin{equation*} (i)\,\ u^\alpha \prec u^\alpha _{\mathbf{J}}\,,\quad \mathbf{J}\neq \mathbf{0},\qquad (ii)\,\ u^\beta _{\mathbf{I}}\prec u^\alpha _{\mathbf{J}}\Longrightarrow D_{\mathbf{K} }u^\beta _{\mathbf{I}}\prec D_{\mathbf{K} }u^\alpha _{\mathbf{J}}\,. \end{equation*}

The leading term in a differential expression is the highest-ranked $u^\alpha _{\mathbf{J}}$ in the expression, and the rank of the expression is the rank of its leading term (see Rust [Reference Rust, Reid and Wittkopf23]). A system of $m$ PDEs, denoted $\mathcal{A}(\mathbf{x},[\mathbf{u}])=0$ , is orthonomic if its components are of the form

(2.1) \begin{equation} \mathcal{A}_\mu =u^{\alpha _\mu }_{\mathbf{J}_\mu }-\omega _\mu (\mathbf{x},[\mathbf{u}]),\qquad \mu =1,\ldots , m, \end{equation}

subject to the following conditions:

1. 1. for each $\mu$ , $u^{\alpha _\mu }_{\mathbf{J}_\mu }$ is ranked higher than every $u^\alpha _{\mathbf{J}}$ that is an argument of $\omega _\mu \,$ ;

2. 2. whenever $\nu \neq \mu$ , the leading term $u^{\alpha _\nu }_{\mathbf{J}_\nu }$ is neither $u^{\alpha _\mu }_{\mathbf{J}_\mu }$ nor a derivative of $u^{\alpha _\mu }_{\mathbf{J}_\mu }$ ;

3. 3. no $u^{\alpha _\mu }_{\mathbf{J}_\mu +\mathbf{K}}$ is an argument of any $\omega _\nu \,$ .

Most systems arising from applications can be written in at least one orthonomic form (commonly several, depending on which ranking is used).

An involutive system has no integrability conditions, and yields a formally well-posed initial-value problem (see Seiler [Reference Seiler24] for details). Every orthonomic system can be completed to an involutive system, denoted ${}^{\mathrm{c}}{{\mathcal{A}}}{}(\mathbf{x},[\mathbf{u}])=0$ , by appending all integrability conditions; Marvan [Reference Marvan14] gives an algorithm for doing this in a finite number of steps. For simplicity, we restrict attention to systems whose involutive completion is orthonomic, subject to the $m$ lowest-ranked components in ${}^{\mathrm{c}}{{\mathcal{A}}}{}$ being the $\mathcal{A}_\mu$ in (2.1). The leading terms in ${}^{\mathrm{c}}{{\mathcal{A}}}{}$ and their derivatives (of all orders) are called principal derivatives; all other $u^\alpha _{\mathbf{J}}$ are called parametric derivatives. Given arbitrary initial data on the set $\mathcal{I}$ of all parametric derivatives, the corresponding power-series solution is constructed by using $\mathcal{A}=0$ and its prolongations to determine the values of all principal derivatives.Footnote ²

Consequently, any function $f(\mathbf{x},[\mathbf{u}])$ may be written, with a slight abuse of notation, as an equivalent function $f(\mathbf{x},\mathcal{I},[\mathcal{A}])$ . This change of coordinates on the infinite jet space uses the variables $\mathbf{D}_{\mathbf{K}}\mathcal{A}_\mu ,\ |\mathbf{K}|\geq 0$ , rather than the principal derivatives. It is extremely useful, as it enables any condition of the form

\begin{equation*} f(\mathbf{x},[\mathbf{u}])=0\quad \text{when}\quad [\mathcal{A}=0] \end{equation*}

(such as the determining equations for symmetries and conservation laws) to be written instead as an equation:

\begin{equation*} f|_0\,:\!=\,f(\mathbf{x},\mathcal{I},[0])=0. \end{equation*}

A syzygy Footnote ³ is a differential relation between variables that amounts to an identity when these variables are written in terms of $(\mathbf{x},\mathbf{u})$ . An orthonomic system of PDEs may have syzygies between the components of $\mathcal{A}$ . (See (3.10) below for a well-known example.) If there are syzygies, the coordinates $\mathbf{D}_{\mathbf{K}}\mathcal{A}_\mu$ have some redundancy.Footnote ⁴ One resolution is to remove spare coordinates. For conservation laws and symmetries, however, it is easier to accept such redundancies and take them into account in the calculations.

2.2 Conservation laws: the basics

A (scalar-valued) conservation law for a system $\mathcal{A}=0$ on $\mathbb{R}^N$ is a divergence expression,

(2.2) \begin{equation} \mathcal{C}(\mathbf{x},[\mathbf{u}])= \text{Div}\,\mathbf{F}= D_iF^i(\mathbf{x},[\mathbf{u}]), \end{equation}

that is zero on all solutions of the PDE:

(2.3) \begin{equation} \mathcal{C}|_0=0. \end{equation}

For orthonomic systems, (2.3) implies that there exist functions $f^{\mu ,\mathbf{J}}$ such that

(2.4) \begin{equation} \mathcal{C}=f^{\mu ,\mathbf{J}}(\mathbf{x},[\mathbf{u}])\,\mathbf{D}_{\mathbf{J}}\mathcal{A}_\mu ; \end{equation}

see Anco [Reference Anco, Melnik, Makarov and Belair1] for details in the involutive case (and Footnote 2). Indeed, many conservation laws have more than one such representation. A conservation law is trivial if either of the following conditions hold:

1. 1. all components $F^i$ are zero when $[\mathcal{A}=0]$ , that is, $F^i|_0=0$ ;

2. 2. $\mathcal{C}=0$ whether or not $[\mathcal{A}=0]$ holds (e.g. when $N=3$ and $\mathbf{F}$ is a ‘total curl’).

More generally, $\mathcal{C}$ is trivial if and only if it is a linear superposition of these two kinds of trivial conservation laws (see Olver [Reference Olver16]), in which case there exist $\widehat {F}^i$ such that

(2.5) \begin{equation} \mathcal{C}=D_i\widehat {F}^i\quad \text{and}\quad \widehat {F}^i|_0=0. \end{equation}

Two conservation laws, $\mathcal{C}_1,\mathcal{C}_2$ , are equivalent if $\mathcal{C}_1\!\!\:-\mathcal{C}_2$ is trivial. Every member of an equivalence class of conservation laws expresses the same information about the set of solutions; so, equivalent conservation laws are generally treated as being identical.

A conservation law $\mathcal{C}$ is in characteristic form if

\begin{equation*} \mathcal{C}=\mathcal{Q}^\mu \mathcal{A}_\mu , \end{equation*}

for some functions $\mathcal{Q}^\mu (\mathbf{x},[\mathbf{u}])$ ; the $m$ -tuple $\mathcal{Q}=(\mathcal{Q}^1,\ldots ,\mathcal{Q}^m)$ is called a multiplier Footnote ⁵ for $\mathcal{C}$ . For a given conservation law, one can integrate any of its representations (2.4) by parts to obtain an equivalent conservation law in characteristic form. This yields functions $\widetilde {F}^i$ satisfying

(2.6) \begin{equation} \mathcal{Q}^\mu \mathcal{A}_\mu =D_i\widetilde {F}^i,\quad \text{where}\quad \mathcal{Q}^\mu =(\!-1)^{|\mathbf{J}|}\,\mathbf{D}_{\mathbf{J}} f^{\mu ,\mathbf{J}}(\mathbf{x},[\mathbf{u}]). \end{equation}

The multiplier $\mathcal{Q}$ is trivial if $\mathcal{Q}|_0=0$ . Two multipliers, $\mathcal{Q}_1$ and $\mathcal{Q}_2$ , are equivalent if $\mathcal{Q}_1\!\!\:-\mathcal{Q}_2$ is trivial. Multipliers are found by solving an overdetermined system of linear PDEs. The solution depends on $[\mathbf{u}]$ and functions of $\mathbf{x}$ that are subject to linear constraints which may or may not be solvable; some or all of these functions may be unconstrained. Anco [Reference Anco, Melnik, Makarov and Belair1] includes full details of how to determine multipliers in a comprehensive review of conservation laws.

2.3 Generalised symmetries

In 1918, Noether [Reference Noether15] famously introduced the idea of generalised symmetries of a system of PDEs (see Olver [Reference Olver17] for a discussion of their significance). For a system $\mathcal{A}=0$ with $M$ components, the $M$ -tuple $\mathbf{Q}=({{Q}}^1(\mathbf{x},[\mathbf{u}]),\ldots ,{{Q}}^M(\mathbf{x},[\mathbf{u}]))$ is the characteristic of a generalised symmetry if the operator

\begin{equation*} X=\mathbf{D}_{\mathbf{J}}\big ({{Q}}^{\alpha }(\mathbf{x},[\mathbf{u}])\big) \frac {{\partial }}{{\partial }\, \mathbf{D}_{\mathbf{J}}u^{\alpha }} \end{equation*}

satisfies the linearised symmetry condition,

(2.7) \begin{equation} X(\mathcal{A}_{\alpha })\big |_0 =0. \end{equation}

Generalised symmetries are not necessarily associated with a Lie group, but the set of all characteristics for a given system is a Lie algebra under the bracket with components

\begin{equation*} [\mathbf{Q}_1,\mathbf{Q}_2]^\alpha =X_1({{Q}}^\alpha _2)-X_2({{Q}}^\alpha _1),\qquad \alpha =1,\ldots ,M. \end{equation*}

In particular, the one-parameter Lie group of point symmetries

\begin{equation*} (\mathbf{x},\mathbf{u})\longmapsto (\widehat {\mathbf{x}}(\mathbf{x},\mathbf{u};\varepsilon),\widehat {\mathbf{u}}(\mathbf{x},\mathbf{u};\varepsilon)), \end{equation*}

defined by

\begin{equation*} \frac {\mathrm{d} \widehat {x}^i}{\mathrm{d}\varepsilon }\,=\xi ^i(\widehat {\mathbf{x}},\widehat {\mathbf{u}}),\qquad \frac {\mathrm{d} \widehat {u}^\alpha }{\mathrm{d}\varepsilon }\,=\eta ^\alpha (\widehat {\mathbf{x}},\widehat {\mathbf{u}}),\qquad \left (\widehat {\mathbf{x}}(\mathbf{x},\mathbf{u};0),\widehat {\mathbf{u}}(\mathbf{x},\mathbf{u};0)\right)=(\mathbf{x},\mathbf{u}), \end{equation*}

has the characteristic whose components are

\begin{equation*} {{Q}}^\alpha =\eta ^\alpha (\mathbf{x},\mathbf{u})-\xi ^i(\mathbf{x},\mathbf{u})u^\alpha _i. \end{equation*}

2.4 Variational symmetries and conservation laws

The formal adjoint of a differential operator $\mathcal{D}$ is the unique differential operator $\mathcal{D}^{\boldsymbol{\dagger }}$ such that

\begin{equation*} F\,\mathcal{D} G - \left (\mathcal{D}^{\boldsymbol{\dagger }} F\right)G \end{equation*}

is a divergence for all smooth functions $F$ and $G$ . In particular,

\begin{equation*} (\mathbf{D}_{\mathbf{J}})^{\boldsymbol{\dagger }}=(\!-\mathbf{D})_{\mathbf{J}}\,:\!=\,(\!-1)^{|\mathbf{J}|}\,\mathbf{D}_{\mathbf{J}}. \end{equation*}

Consequently, the Euler–Lagrange operator with respect to any variable $v$ that depends on $\mathbf{x}$ is

\begin{equation*} \mathbf{E}_v=(\!-\mathbf{D})_{\mathbf{J}}\,\frac {{\partial } }{{\partial } v_{\mathbf{J}}}\,,\qquad v_{\mathbf{J}}\,:\!=\,\mathbf{D}_{\mathbf{J}}v, \end{equation*}

where the total derivatives $\mathbf{D}_{\mathbf{J}}$ now include all dependent variables, including $v$ . It is well known that a function $f(\mathbf{x},[\mathbf{u}])$ is a divergence if and only if

\begin{equation*} \mathbf{E}_{u^\alpha }\{f(\mathbf{x},[\mathbf{u}])\}=0,\qquad \alpha =1,\ldots , M. \end{equation*}

A useful generalisation of this result applies to smooth functions that depend on $\mathbf{x}$ , $[v]$ and a set, $\mathbf{z}$ , that consists of other (subsidiary) dependent variables and their derivatives, as follows.

Lemma 2.1. Let $f$ and $F^i,\ i=1,\ldots , N$ , be arbitrary smooth functions of $\mathbf{x}$ , a set of dependent variables, $v$ and their derivatives and a set of subsidiary dependent variables, $\mathbf{z}$ . Suppose that no syzygies exist between the variables $(\mathbf{x},\mathbf{z},[v])$ . Suppose also that for all $i$ , each subsidiary $z^a$ on which $F^i(\mathbf{x},\mathbf{z},[v])$ depends satisfies the condition that $D_iz^a$ is a function of $(\mathbf{x},\mathbf{z})$ only. Then, $\mathbf{E}_v\!\left (D_iF^i\right)=0$ . If $\mathbf{E}_{v}\big \{f(\mathbf{x},\mathbf{z},[v])\big \}=0$ , then $f(\mathbf{x},\mathbf{z},[v])-f(\mathbf{x},\mathbf{z},[0])$ is a divergence.

Proof. Suppose that $D_iz^a$ is independent of $[v]$ , for all $z^a$ occurring in $F^i$ . Then,

\begin{align*} \mathbf{E}_v\!\left (D_iF^i(\mathbf{x},\mathbf{z},[v])\right) &=(\!-\mathbf{D})_{\mathbf{J}}\left \{D_i\!\left (\frac {{\partial } F^i}{{\partial } v_{\mathbf{J}}}\right)\!+\frac {{\partial } D_iz^a}{{\partial } v_{\mathbf{J}}} \,\frac {{\partial } F^i}{{\partial } z^a}+\,\frac {{\partial } v_{\mathbf{K} i}}{{\partial } v_{\mathbf{J}}} \,\frac {{\partial } F^i}{{\partial } v_{\mathbf{K}}}\right \}\\ &=(\!-\mathbf{D})_{\mathbf{J}}\left \{D_i\!\left (\frac {{\partial } F^i}{{\partial } v_{\mathbf{J}}}\right)+\,\frac {{\partial } v_{\mathbf{K} i}}{{\partial } v_{\mathbf{J}}} \,\frac {{\partial } F^i}{{\partial } v_{\mathbf{K}}}\right \}\\ &=0. \end{align*}

(Syzygies invalidate the conclusion ‘ $=0$ ’.) Now, suppose that $\mathbf{E}_{v}\big \{f(\mathbf{x},\mathbf{z},[v])\big \}=0$ . Then,

\begin{align*} f(\mathbf{x},\mathbf{z},[v])-f(\mathbf{x},\mathbf{z},[0])&=\int _{t=0}^1\frac {\mathrm{d} f(\mathbf{x},\mathbf{z},[tv])}{\mathrm{d} t}-v\left (\mathbf{E}_{v}\big \{f(\mathbf{x},\mathbf{z},[v])\big \} \right)\!\big |_{v\mapsto tv}\,\mathrm{d} t\\ &=\int _{t=0}^1\left \{v_{\mathbf{J}}\,\frac {{\partial } f(\mathbf{x},\mathbf{z},[v])}{{\partial } v_{\mathbf{J}}}-v(\!-\mathbf{D})_{\mathbf{J}}\,\frac {{\partial } v_{\mathbf{J}}}-v(\!-\mathbf{D})_{\mathbf{J}}\,\frac {{\partial } f(\mathbf{x},\mathbf{z},[v])}{{\partial } v_{\mathbf{J}}}\right \}\bigg |_{v\mapsto tv}\frac {\mathrm{d} t}{t}\,. \end{align*}

The expression in braces is a divergence; so, $f(\mathbf{x},\mathbf{z},[v])-f(\mathbf{x},\mathbf{z},[0])$ is a divergence.

For the rest of this subsection, let $\mathcal{A}=0$ be the system of Euler–Lagrange equations,

(2.8) \begin{equation} \mathcal{A}_\alpha \,:\!=\,{\mathbf{E}}_{u^\alpha }\{L(\mathbf{x},[\mathbf{u}])\}=0,\qquad \alpha =1,\ldots ,M, \end{equation}

arising from the variational problem $\delta \mathscr{L}=0$ , where

\begin{equation*} \mathscr{L}=\int L(\mathbf{x},[\mathbf{u}]) \,\mathrm{d}\mathbf{x} \end{equation*}

is the action. In view of our assumption that the domain is $\mathbb{R}^N$ , we work formally, assuming that all integrals converge. We begin with a brief outline of variational symmetries; for more details, including the modifications needed for domains with a boundary, see Olver [Reference Olver16].

A generalised symmetry is variational if, for all sufficiently small $|\varepsilon |$ , the mapping

\begin{equation*} (\mathbf{x},\,u^\alpha _{\mathbf{J}})\longmapsto (\mathbf{x},\,u^\alpha _{\mathbf{J}}+\varepsilon \mathbf{D}_{\mathbf{J}} {{Q}}^\alpha) \end{equation*}

leaves the action unchanged, to first order in $\varepsilon$ . Consequently, the Euler–Lagrange equations are unchanged,Footnote ⁶ to first order. As

\begin{equation*} L\mapsto L+\varepsilon X(L) + O(\varepsilon ^2), \end{equation*}

it follows that $X(L)$ belongs to the kernel of all $\mathbf{E}_{u^\alpha }$ ; so, it is a divergence. Therefore, $\mathbf{Q}$ is a characteristic of generalised variational symmetries if and only if there exist functions $P^i(\mathbf{x},[\mathbf{u}])$ such that

(2.9) \begin{equation} X(L) = \mathbf{D}_{\mathbf{J}}{{Q}}^{\alpha }\, \frac {\partial L}{{\partial }\, \mathbf{D}_{\mathbf{J}}u^{\alpha }} =D_i P^i. \end{equation}

This can be integrated by parts to obtain a conservation law in characteristic form,

(2.10) \begin{equation} \qquad {{Q}}^{\alpha }{\mathbf{E}}_{\alpha }(L) = D_i F^i(\mathbf{x},[\mathbf{u}]), \end{equation}

where $D_i(F^i-P^i)$ is the divergence arising from the integration. Furthermore, given any conservation law in characteristic form, the argument above can be reversed to go from (2.10) to (2.9) (with $\mathcal{Q}^\alpha$ replacing ${{Q}}^\alpha$ ). This leads to Noether’s (First) Theorem from her 1918 paper.

Theorem 2.2 (Noether $1$ ). A function $\mathbf{Q}$ is a generalised variational symmetry characteristic for a system of Euler–Lagrange equations if and only if it is a multiplier of a conservation law.

Noether’s Second Theorem deals with families of characteristics that depend on a completely arbitrary function ${\mathtt {g}}(\mathbf{x})$ and its derivatives.

Theorem 2.3 (Noether $2$ ). An Euler–Lagrange system, $\mathbf{E}_{u^\alpha }(L)=0$ , has a family of variational symmetry characteristics $\mathbf{Q}(\mathbf{x},[\mathbf{u}],[{\mathtt {g}}])$ that are linear homogeneous in an arbitrary function ${\mathtt {g}}(\mathbf{x})$ if and only if there is a syzygy between the components $\mathbf{E}_{u^\alpha }(L)$ , that is, if and only if there exist differential operators $\mathcal{D}^\alpha$ (independent of $\mathtt {g}$ ) such that

(2.11) \begin{equation} \mathcal{D}^{\alpha } \mathbf{E}_{u^\alpha }(L)= 0. \end{equation}

Specifically, the family of characteristics is obtained from this syzygy by multiplying (2.11) by $\mathtt {g}$ , then integrating by parts to obtain an expression of the form (2.10), whose right-hand side is (automatically) a trivial conservation law. Recently, Olver has proved that the arbitrary function $\mathtt {g}$ may also depend on $[\mathbf{u}]$ (see Olver [Reference Olver18] for details). Noether’s Second Theorem immediately generalises to families of characteristics that depend on more than one arbitrary function, each of which corresponds to an independent syzygy.

3. Conservation laws that depend on functions of $\mathbf{x}$

From here on, we consider a given PDE system $\mathcal{A}=0$ that is not necessarily variational. The set $C_{\:\!\!\mathcal{A}}$ of equivalence classes of conservation laws is a vector space. If $C_{\:\!\!\mathcal{A}}$ is of countable dimension (whether finite or not), it has a basis of equivalence classes that can be indexed by $k\in \mathbf{N}$ . For each $k$ , let $\mathcal{C}_k$ be a non-zero conservation law in the $k^{\text{th}}$ equivalence class. Then, every conservation law is equivalent to $\mathcal{C}=c^k\mathcal{C}_k$ for some constants $c^k$ . However, for many well-known systems, $C_{\:\!\!\mathcal{A}}$ is uncountable, because there exist conservation laws in a family $\mathcal{C}(\mathbf{x},[\mathbf{u}],[{\mathtt {\mathbf{g}}}])$ that is linear homogeneous in ${\mathtt {\mathbf{g}}}=({\mathtt {g}}^1(\mathbf{x}),{\mathtt {g}}^2(\mathbf{x}),\ldots)$ . The functions ${\mathtt {g}}^r$ may be free (entirely arbitrary) or arbitrary subject to some linear differential constraints,

(3.1) \begin{equation} \mathscr{D}^{\;l}_r{\mathtt {g}}^{r}=0,\qquad l=1,2,\ldots \, . \end{equation}

Here, each $\mathscr{D}^{\;l}_r$ is a linear differential operator whose coefficients depend on $\mathbf{x}$ only; total derivatives $D_i$ are used, even though ${\mathtt {g}}^r$ does not depend on $[\mathbf{u}]$ . The set of constraints (3.1) is complete if:

• • it fully specifies $\mathtt {\mathbf{g}}$ , so that each ${\mathtt {g}}^r$ is an arbitrary function of $\mathbf{x}$ subject only to (3.1);

• • it has no additional integrability conditions.

Note: If the dimension of $C_{\:\!\!\mathcal{A}}$ is countable, every conservation law $\mathcal{C}$ is a member of the family $\mathcal{C}={\mathtt {g}}^k\mathcal{C}_k(\mathbf{x},[\mathbf{u}])$ that is subject to the constraints $D_i{\mathtt {g}}^k=0$ for all $i$ and $k$ . So, $C_{\:\!\!\mathcal{A}}$ having countable dimension can be regarded as a special case of a family of conservation laws depending linearly on functions that are arbitrary subject to a given set of linear constraints.

Multipliers that depend on functions $\mathtt {\mathbf{g}}$ can be used to obtain syzygies and/or simplify the conservation laws of the given system of PDEs.

Theorem 3.1. Suppose that $\mathcal{A}=0$ has a family of conservation laws $\mathcal{C}$ that is linear homogeneous in $\mathtt {\mathbf{g}}$ , which is subject to the complete set of constraints ( 3.1 ). Then, there exists $\boldsymbol{\lambda }=(\lambda _1(\mathbf{x},[\mathbf{u}]),\lambda _2(\mathbf{x},[\mathbf{u}]),\ldots)$ such that

(3.2) \begin{equation} (\!-\mathbf{D})_{\mathbf{J}}\left (\frac {{\partial }\, \mathcal{C}(\mathbf{x},[\mathbf{u}],[{\mathtt {\mathbf{g}}}])}{{\partial } {\mathtt {g}}^r_{\mathbf{J}}}\right)+\big (\mathscr{D}^{\;l}_r\big)^{\!\boldsymbol{\dagger }}\lambda _l=0,\qquad r=1,2,\ldots . \end{equation}

For each set of differential operators $\mathscr{D}^{\;r}$ such that $\mathscr{D}^{\;r}(\mathscr{D}^{\;l}_r)^{\!\boldsymbol{\dagger }}\lambda _l=0$ , there is a corresponding syzygy,

(3.3) \begin{equation} \mathscr{D}^{\;r}(\!-\mathbf{D})_{\mathbf{J}}\left (\frac {{\partial }\, \mathcal{C}(\mathbf{x},[\mathbf{u}],[{\mathtt {\mathbf{g}}}])}{{\partial } {\mathtt {g}}^r_{\mathbf{J}}}\right)=0. \end{equation}

Provided that not all ${\mathtt {g}}^r$ are free, the family $\mathcal{C}$ is equivalent to the family of conservation laws obtained by substituting any solution $\boldsymbol{\lambda }$ of ( 3.2 ) into

(3.4) \begin{equation} \mathcal{C}_{\boldsymbol{\lambda }}\,:\!=\,\lambda _l\mathscr{D}^{\;l}_r{\mathtt {g}}^r-{\mathtt {g}}^r\big (\mathscr{D}^{\;l}_r\big)^{\!\boldsymbol{\dagger }}\lambda _l. \end{equation}

Proof. Suppose that $\mathcal{A}=0$ has such a family of conservation laws. If any of the functions ${\mathtt {g}}^r$ are free, Lemma 2.1 applies with $v={\mathtt {g}}^r$ , giving

(3.5) \begin{equation} \mathbf{E}_{{\mathtt {g}}^r}\!\left \{\mathcal{C}(\mathbf{x},[\mathbf{u}],[{\mathtt {\mathbf{g}}}])\right \}=0. \end{equation}

For all other $r$ , use Lagrange multipliers, $\lambda _l$ , to enforce the constraints, giving

(3.6) \begin{equation} \mathbf{E}_{{\mathtt {g}}^r}\!\left \{\mathcal{C}(\mathbf{x},[\mathbf{u}],[{\mathtt {\mathbf{g}}}])+\lambda _l\:\!\mathscr{D}^{\;l}_s{\mathtt {g}}^s\right \}=0. \end{equation}

This amounts to (3.2), from which the syzygies (3.3) are constructed. As $\lambda _l\mathscr{D}_s^{\;l}=0$ when ${\mathtt {g}}^s$ is free, (3.5) can be treated as a special case of the general construction. If not all $\mathscr{D}^{\;l}_r$ are zero, there exists a solution $\boldsymbol{\lambda }(\mathbf{x},[\mathbf{u}])$ of (3.2).

The conservation law $\mathcal{C}$ differs from its characteristic form $\mathcal{Q}^\mu \mathcal{A}_\mu$ by a divergence. Lagrange multipliers enable the functions ${\mathtt {g}}^r$ to be treated as if they were unrelated so, by Lemma 2.1,

(3.7) \begin{equation} \mathbf{E}_{{\mathtt {g}}^r}\{\mathcal{C}+\lambda _l\:\!\mathscr{D}^{\;l}_s{\mathtt {g}}^s\}=\mathbf{E}_{{\mathtt {g}}^r}\{\mathcal{Q}^\mu \mathcal{A}_\mu +\lambda _l\:\!\mathscr{D}^{\;l}_s{\mathtt {g}}^s\}=(\!-\mathbf{D})_{\mathbf{J}}\left (\frac {{\partial }\, \mathcal{Q}^\mu (\mathbf{x},[\mathbf{u}],[{\mathtt {\mathbf{g}}}])}{{\partial } {\mathtt {g}}^r_{\mathbf{J}}}\,\mathcal{A}_\mu \right)+\big (\mathscr{D}^{\;l}_r\big)^{\!\boldsymbol{\dagger }}\lambda _l. \end{equation}

As $\mathcal{Q}$ is linear homogeneous in $\mathtt {\mathbf{g}}$ ,

\begin{equation*} \mathcal{Q}^\mu \mathcal{A}_\mu -\mathcal{C}_{\boldsymbol{\lambda }}={\mathtt {g}}^r_{\mathbf{J}}\, \frac {{\partial } \mathcal{Q}^\mu }{{\partial } {\mathtt {g}}^r_{\mathbf{J}}}\,\mathcal{A}_\mu -{\mathtt {g}}^r(\!-\mathbf{D})_{\mathbf{J}}\left (\frac {{\partial } \mathcal{Q}^\mu }{{\partial } {\mathtt {g}}^r_{\mathbf{J}}}\,\mathcal{A}_\mu \right), \end{equation*}

for all $\mathtt {\mathbf{g}}$ that satisfy the constraints. This is a trivial conservation law; thus, so is $\mathcal{C}-\mathcal{C}_{\boldsymbol{\lambda }}$ .

Finally, suppose that $\boldsymbol{\lambda }^{1}$ and $\boldsymbol{\lambda }^{2}$ both satisfy (3.2) and let $\boldsymbol{\lambda }=\boldsymbol{\lambda }^{1}-\boldsymbol{\lambda }^{2}$ . Then, $\big (\mathscr{D}^{\;l}_r\big)^{\!\boldsymbol{\dagger }}\lambda _l=0$ , and so $\mathcal{C}_{\boldsymbol{\lambda }}=\mathcal{C}_{\boldsymbol{\lambda }^{1}}-\mathcal{C}_{\boldsymbol{\lambda }^{2}}$ is zero for all $\mathtt {\mathbf{g}}$ that satisfy the given constraints. Therefore, every solution of (3.2) yields a family of conservation laws (3.4) that is equivalent to $\mathcal{C}$ .

It is well known that if $\ell u=0$ is a linear homogeneous PDE (where $\ell$ is a linear differential operator) and ${\mathtt {g}}(\mathbf{x})$ is a solution of $\ell ^{\boldsymbol{\dagger }}{\mathtt {g}}=0$ , then

(3.8) \begin{equation} \mathcal{C}={\mathtt {g}} \ell u-u\ell ^{\boldsymbol{\dagger }}{\mathtt {g}} \end{equation}

is a conservation law. This arises from Theorem3.1 by setting $\mathcal{A}_1=\ell u$ and $\mathcal{Q}^1={\mathtt {g}}^1={\mathtt {g}}(\mathbf{x})$ ; the condition $\mathbf{E}_u(\mathcal{Q}^1\mathcal{A}_1)=0$ gives the constraint $\ell ^{\boldsymbol{\dagger }}{\mathtt {g}}=0$ ; so, $\mathscr{D}_1^1=\ell ^{\boldsymbol{\dagger }}$ and hence (3.2) has the solution $\lambda _1=-u$ . Theorem3.1 extends this result to non-linear systems of PDEs.

As Theorem3.1 implies, there are systems of PDEs and constraints for which it is possible to eliminate Lagrange multipliers from (3.2) and its differential consequences to obtain syzygies.

Example 3.2. In Cartesian coordinates, the Euler equation for a constant-density two-dimensional incompressible potential flow with velocity $\nabla \phi$ and pressure $p$ has the components

\begin{equation*} \mathcal{A}_1=\phi _{xt}+\phi _x\phi _{xx}+\phi _y\phi _{xy}+p_x,\qquad \mathcal{A}_2=\phi _{yt}+\phi _x\phi _{xy}+\phi _y\phi _{yy}+p_y,\qquad \mathcal{A}_3=\phi _{xx}+\phi _{yy}\,. \end{equation*}

This system has an integrability condition Footnote ⁷ that determines the Laplacian of the pressure in terms of $[\phi ]$ . All multipliers that are independent of $([\phi ],[p])$ are of the form $\mathcal{Q}^\mu ={\mathtt {g}}^{\mu }(x,y,t)$ , subject to the following constraints:

\begin{equation*} D_x{\mathtt {g}}^1+D_y{\mathtt {g}}^2=0,\qquad D_x^2{\mathtt {g}}^3+D_y^2{\mathtt {g}}^3=0. \end{equation*}

Applying Theorem 3.1 gives the conditions

(3.9) \begin{equation} \mathcal{A}_1=D_x\lambda _1,\qquad \mathcal{A}_2=D_y\lambda _1,\qquad \mathcal{A}_3=-D_x^2\lambda _2-D_y^2\lambda _2, \end{equation}

which have a solution $(\lambda _1,\lambda _2)=(H,-\phi)$ , where $H=\phi _t+(\phi _x^2+\phi _y^2)/2+p$ is the Bernoulli function. The corresponding family of conservation laws is $\mathcal{C}_{\boldsymbol{\lambda }}=\mathcal{C}_1+\mathcal{C}_2$ , where

\begin{equation*} \mathcal{C}_1=D_x\left({\mathtt {g}}^1H\right)+D_y\left({\mathtt {g}}^2H\right),\qquad \mathcal{C}_2=D_x\left({\mathtt {g}}^3\phi _x-{\mathtt {g}}^3_x\phi \right)+D_y\left({\mathtt {g}}^3\phi _y-{\mathtt {g}}^3_y\phi \right). \end{equation*}

Note that $\lambda _1$ can be eliminated from ( 3.9 ), yielding the syzygy

(3.10) \begin{equation} D_x\mathcal{A}_2-D_y\mathcal{A}_1=0. \end{equation}

The conservation law $\mathcal{C}_1$ is trivial, because the solution of the first constraint is $({\mathtt {g}}^1,{\mathtt {g}}^2)=({\mathtt {g}}_y,-{\mathtt {g}}_x)$ , where the arbitrary function ${\mathtt {g}}(x,y,t)$ can be regarded as the Lagrange multiplier for the syzygy. Consequently,

\begin{equation*} \mathcal{C}_1=D_x\left (D_y({\mathtt {g}} H)-{\mathtt {g}}\mathcal{A}_2\right)+D_y\left (\!-D_x({\mathtt {g}} H)+{\mathtt {g}}\mathcal{A}_1\right)=D_x\left (-{\mathtt {g}}\mathcal{A}_2\right)+D_y\left ({\mathtt {g}}\mathcal{A}_1\right). \end{equation*}

By contrast, $\mathcal{C}_2$ is non-trivial for ${\mathtt {g}}^3\neq 0$ ; indeed, it is the conservation law ( 3.8 ) for Laplace’s equation.

In particular, Corollary 3.3 applies whenever $\mathtt {\mathbf{g}}$ is an arbitrary function of some (but not all) independent variables.Footnote ⁸ The best-known examples with this property are scalar PDEs with first integrals, such as the Liouville equation and other Darboux integrable scalar hyperbolic PDEs. However, the corollary applies equally to multi-component systems whose multipliers depend one or more arbitrary functions of $N-1$ variables.

Example 3.4. Consider the Liouville-type system $\mathcal{A}=0$ defined by

\begin{equation*} \mathcal{A}_1=u_{xt}-e^{2u-v},\qquad \mathcal{A}_2=v_{xt}-e^{2v-u}. \end{equation*}

This has a family of multipliers that depend on $(u,v,u_x,v_x)$ and ${\mathtt {g}}^1(x,t)$ , with components

\begin{equation*} \mathcal{Q}^1={\mathtt {g}}^1(2u_x-v_x)+D_x{\mathtt {g}}^1,\qquad \mathcal{Q}^2={\mathtt {g}}^1(2v_x-u_x)+D_x{\mathtt {g}}^1, \end{equation*}

subject to the constraint $D_t{\mathtt {g}}^1=0$ . The corresponding conservation law in characteristic form is

\begin{equation*} \mathcal{C}=D_x\left \{-\,{\mathtt {g}}^1\!\left (e^{2u-v}+e^{2v-u}\right)\right \}+D_t\left \{{\mathtt {g}}^1\!\left (u_x^2-u_xv_x+v_x^2\right)+(D_x{\mathtt {g}}^1)(u_x+v_x)\right \}. \end{equation*}

The condition ( 3.2 ) amounts to

\begin{equation*} (2u_x-v_x-D_x)\mathcal{A}_1+(2v_x-u_x-D_x)\mathcal{A}_2-D_t\lambda _1=0, \end{equation*}

which has a solution

\begin{equation*} \lambda _1=u_x^2-u_xv_x+v_x^2-u_{xx}-v_{xx}\,. \end{equation*}

So, this family of multipliers leads to the family of conservation laws $D_t({\mathtt {g}}^1 \lambda _1)$ for all ${\mathtt {g}}^1$ that depend on $x$ only; in other words, $\lambda _1$ is a first integral.

Similarly, for the multiplier with components

\begin{equation*} \mathcal{Q}^1={\mathtt {g}}^2(2u_xv_x-v_x^2+v_{xx})+2(D_x{\mathtt {g}}^2)u_x+D_x^2{\mathtt {g}}^2,\qquad \mathcal{Q}^2={\mathtt {g}}^2(u_x^2-2u_xv_x-u_{xx})-(D_x{\mathtt {g}}^2)u_x\,, \end{equation*}

subject to $D_t{\mathtt {g}}^2=0$ , the condition ( 3.2 ) leads to another first integral,

\begin{equation*} \lambda _2=u_x(v_x^2-u_xv_x+2u_{xx}-v_{xx})-u_{xxx}\,. \end{equation*}

Unlike $\lambda _1$ , the function $\lambda _2$ is not symmetric under the discrete symmetry $(u,v)\mapsto (v,u)$ . This symmetry produces the first integral $D_x\lambda _1-\lambda _2$ . The discrete symmetry $(x,t)\mapsto (t,x)$ generates further first integrals from $\lambda _1$ and $\lambda _2$ .

More generally, the constraint need not be of the form $D_i{\mathtt {g}}=0$ for a first integral to arise for a PDE system with two independent variables. The constraint $\mathscr{D}{\mathtt {g}}=0$ is sufficient, for any first-order differential operator $\mathscr{D}$ whose coefficients depend on $\mathbf{x}$ only.

Example 3.5. The shallow water equations (with constant Coriolis parameter $f$ ) describe the position $(x,y)$ of a fluid particle in terms of label-space variables $(m^1,m^2)$ and time $t$ . Denoting differentiation with respect to $m^i$ by the subscript $i$ , this PDE system $\mathcal{A}=0$ has two components, with

\begin{align*} \mathcal{A}_1=&\ x_{tt}-fy_t-D_1(y_1e'(\tau))+D_2(y_1e'(\tau)),\\ \mathcal{A}_2=&\ y_{tt}+fx_t+D_1(x_2e'(\tau))-D_2(x_1e'(\tau)), \end{align*}

where $\tau =x_1y_2-x_2y_1$ is the reciprocal of the fluid depth and $e(\tau)$ is the internal energy of the fluid. This system has a family of conservation laws that depend on a function $\mathtt {g}$ which is arbitrary subject to the constraint $D_t({\mathtt {g}})=0$ , namely

\begin{align*} \mathcal{C}=&\ D_t\!\left \{({\mathtt {g}}_2x_1-{\mathtt {g}}_1x_2)x_t+(y_t+fx)({\mathtt {g}}_2y_1-{\mathtt {g}}_1y_2)\right \}+D_1\!\left \{{\mathtt {g}}_2\!\left (e(\tau)-\tau e'(\tau)-\tfrac {1}{2} (x_t^2+y_t^2)-fxy_t\right)\right \}\\ &+D_2\!\left \{{\mathtt {g}}_1\!\left (\tau e'(\tau)-e(\tau)+\tfrac {1}{2} (x_t^2+y_t^2)+fxy_t\right)\right \}. \end{align*}

The special cases ${\mathtt {g}}=-m^1$ and ${\mathtt {g}}=m^2$ give a pair of conservation laws for Lagrangian momentum (see Hydon [7]). Applying Theorem 3.1 gives (after simplification)

\begin{equation*} (x_2D_1-x_1D_2)\mathcal{A}_1+(y_2D_1-y_1D_2)\mathcal{A}_2-D_t\lambda =0, \end{equation*}

which is solved by

\begin{equation*} \lambda =f\tau +x_2x_{1t}-x_1x_{2t}+y_2y_{1t}-y_1y_{2t}. \end{equation*}

The conservation law $\mathcal{C}$ is equivalent to

\begin{equation*} C_\lambda =D_t\{{\mathtt {g}} \lambda \}, \end{equation*}

which expresses conservation of the potential vorticity, $\lambda$ .

Another application of the results in this section occurs where symbolic algebra has been used to derive conservation laws (see e.g. Poole and Hereman [Reference Poole and Hereman20] and Wolf [Reference Wolf27]); these are not necessarily in their simplest equivalent form. Once a family of conservation laws that depend on a function is known, Theorem3.1 may be used with Corollary 3.3 to find an equivalent (perhaps simpler) form.

Example 3.6. For the KP equation, written as the system

\begin{equation*} v_x-u_{yy}=0,\qquad v=u_{t}+2uu_{x}+u_{xxx}\,, \end{equation*}

symbolic methods yield the following family of conservation laws with $\mathtt {g}$ a function of $t$ only:

\begin{equation*} \mathcal{C}=D_t\{{\mathtt {g}} yu\}+D_x\left \{{\mathtt {g}} yu_{xx}+{\mathtt {g}} yu^2-\big (\tfrac {1}{6}{\mathtt {g}}_{t}y^3+{\mathtt {g}} xy\big)v\right \}+D_y\left \{\big (\tfrac {1}{6}{\mathtt {g}}_{t}y^3+{\mathtt {g}} xy\big)u_{y}-\tfrac {1}{2}{\mathtt {g}}_{t}y^2u-{\mathtt {g}} xu\right \}\!. \end{equation*}

Applying Theorem 3.1 with the constraints $D_x{\mathtt {g}}=0$ , $D_y{\mathtt {g}}=0$ , gives the equivalent family of conservation laws,

\begin{equation*} \mathcal{C}_\lambda =D_x\left \{{\mathtt {g}}\big ( yu_{xx}+yu^2+\tfrac {1}{6}y^3v_t-xyv\big)\right \}+D_y\left \{{\mathtt {g}}\big (-\tfrac {1}{6}y^3u_{ty}+xyu_{y}+\tfrac {1}{2}y^2u_{t}-xu\big)\right\}\!, \end{equation*}

which is a little simpler than $\mathcal{C}$ , as it has one fewer term and no $D_t$ component. For higher-order conservation laws with many terms, greater simplification can occur.

Note. When the vector space $C_{\mathcal{A}}$ of equivalence classes of conservation laws has countable dimension, with a basis $\{\mathcal{C}_k=D_iF_k^i(\mathbf{x},[\mathbf{u}]): k=1,2,\ldots ,\}$ ), one can represent the set of all conservation laws (up to equivalence) by

\begin{equation*} \mathcal{C}={\mathtt {g}}^k\mathcal{C}_k,\quad \text{subject to}\quad D_i{\mathtt {g}}^k=0. \end{equation*}

With a slight variation in notation, let $\lambda ^i_k$ be the Lagrange multiplier corresponding to the constraint $D_i{\mathtt {g}}^k=0$ . Then, (3.2) amounts to

\begin{equation*} \mathbf{E}_{{\mathtt {g}}^j}(\mathcal{C})=D_i\lambda ^i_j\,, \end{equation*}

which has the solution $\lambda ^i_j=F^i_j(\mathbf{x},[\mathbf{u}])$ ; the resulting conservation law is $\mathcal{C}_{\boldsymbol{\lambda }}=\mathcal{C}$ . So, the approach taken above applies equally whether $C_{\mathcal{A}}$ is finite- or infinite-dimensional.

4. Hodograph transformations

This section examines what happens when a system of PDEs has a family of conservation laws that depend on arbitrary functions which involve dependent variables. We restrict attention to systems for which there exists a hodograph transformation that gives all such variables the role of independent variables. Let $\widetilde {x}^i(\mathbf{x},\mathbf{u}),\ i=1,\ldots N$ be the new independent variables after such a transformation, and let $\widetilde {u}^\alpha (\mathbf{x},\mathbf{u}),\ \alpha =1,\ldots ,M$ be the new dependent variables. For the transformation to be valid, it is necessary that the Jacobian determinant, $\mathcal{J}=\text{det}(D_j \widetilde {x}^i)$ , is non-zero. Let $\widetilde {D}_i$ denote the total derivative with respect to $\widetilde {x}^i$ in the new variables. Then, the change of variables rule for a total divergence gives the following useful result.

Consequently, the results of Section 3 can be applied immediately to the transformed PDE system.

Example 4.2. Consider the time-dependent problem of flow by mean curvature via its level-set formulation,

(4.1) \begin{equation} |\nabla u| \nabla \cdot \left (\frac {\nabla u}{|\nabla u|} \right)-u_t=0. \end{equation}

We will work in two spatial dimensions, though the following conservation-law reduction generalises readily to higher dimensions and to other forms of interfacial dynamics. The normal velocity of each level set of $u(x,y,t)$ is given by the negative of its curvature (so that closed curves disappear in finite time). This geometrical content of ( 4.1 ), whereby there is no coupling between different level sets, is reflected in its having three obvious families of symmetries,

(4.2) \begin{equation} u\longmapsto U(u),\quad U'(u)\neq 0,\qquad x\longmapsto x + X(u),\qquad y \longmapsto y + Y(u), \end{equation}

each involving an arbitrary function of $u$ . Equation ( 4.1 ) admits every multiplier of the form $\mathcal{Q}={\mathtt {g}}(u)$ ; in neighbourhoods where $u_x\neq 0$ , the resulting conservation law is

(4.3) \begin{equation} D_x\{-{\mathtt {g}}(u)u_y\tan ^{-1}(u_y/u_x)\}+D_y\{{\mathtt {g}}(u)u_x\tan ^{-1}(u_y/u_x)\}+D_t\left \{-\int {\mathtt {g}}(u)\,\mathrm{d} u\right \}=0. \end{equation}

Proceeding as above in applying a hodograph transformation with $u$ as an independent variable and $x=x(u,y,t)$ as a dependent variable, ( 4.3 ) reduces to the product of ${\mathtt {g}}(u)$ and the conservation law

(4.4) \begin{equation} D_y\left \{-\tan ^{-1}(x_y)\right \}+D_t\{x\}=0. \end{equation}

A similar reduction using $y$ as a dependent variable applies in neighbourhoods where $u_y\neq 0$ . (One could also use $t$ a dependent variable, but the resulting conservation law is not much simpler than ( 4.3 ).)

So, conservation-law reduction simplifies the time-dependent two-dimensional mean curvature flow to the non-linear filtration equation

(4.5) \begin{equation} x_t=\frac {x_{yy}}{1+x_y^2}\,; \end{equation}

see Ibragimov [9] for symmetries and some exact solutions of ( 4.5 ). The five Lie point symmetries in Ibragimov [9] amount to translations in $x, y$ and $t$ , a scaling, and rotations in the $(x,y)$ -plane. Each of these corresponds to a family of symmetries of ( 4.1 ) that is obtained by replacing each group parameter by an arbitrary function of $u$ . For instance, the scaling invariance of ( 4.5 ) amounts to the family of symmetries

\begin{equation*} x \longmapsto h(u)x,\qquad y \longmapsto h(u)y,\qquad t\longmapsto h^{2}(u)t, \end{equation*}

of ( 4.1 ) for arbitrary non-zero $h(u)$ , which is perhaps not obvious a priori from ( 4.1 )Footnote ⁹ . The first family of symmetries in ( 4.2 ) becomes trivial in the reduced equation, as $u$ appears only parametrically in ( 4.5 ) (so that the dimensionality is lowered).

Equation ( 4.5 ) is of course familiar in the interfacial dynamics context: parametrising the level set of interest in the form $x=f(y,t)$ by setting $u=x-f(y,t)$ leads at once to ( 4.5 ) with $x$ replaced by $-f$ ; the reduction to ( 4.5 ) is in this sense obvious.

Among the many natural generalisations of ( 4.1 ), here we mention only the anisotropic case

\begin{equation*} u_t=\Phi '\left (\tan ^{-1}\left (u_y/u_x\right)\right)|\nabla u| \,\nabla \!\cdot \!\left (\frac {\nabla u}{|\nabla u|} \right), \end{equation*}

that similarly reduces to

\begin{equation*} D_y\left \{\Phi (-\tan ^{-1}(x_y))\right \}+D_t(x)=0. \end{equation*}

5. Multiple reduction using conservation laws and symmetries

For clarity, we now consider scalar PDEs $\mathcal{A}=0$ with two independent variables, $(x,t)$ . Suppose that a given PDE has a family of conservation laws whose multipliers depend linearly on an arbitrary function ${\mathtt {g}}(x,t)$ that is subject to a constraint of the form

\begin{equation*} a(x,t)D_x{\mathtt {g}}+b(x,t)D_t{\mathtt {g}}=0. \end{equation*}

Using the method of characteristics to change variables if necessary, suppose without loss of generality that $D_x{\mathtt {g}}=0$ . Corollary 3.3 implies that (up to equivalence) the corresponding family of conservation laws (3.4) is of the form

\begin{equation*} \mathcal{C}\,:\!=\,D_x\{{\mathtt {g}}(t)\lambda (x,t,[u])\}=0. \end{equation*}

Consequently, $\lambda (x,t,[u])$ is a first integral and so the PDE reduces to

(5.1) \begin{equation} \lambda (x,t,[u])=f(t), \end{equation}

where $f$ is arbitrary. Symmetries of $\mathcal{A}=0$ map the set of solutions to itself, so they are equivalence transformations of (5.1), that is, they may change $f$ . Thus, a known Lie group of symmetries can be used to partition the reduced PDE (5.1) into equivalence classes. Commonly, the symmetry group depends on at least one arbitrary function, $h(t)$ . In this case, typically, the number of equivalence classes is small and each class has a member with $f$ constant. At the other extreme, some non-trivial symmetries of $\mathcal{A}=0$ may become trivial symmetries of the reduced equation (5.1); such symmetries do not simplify (5.1).

Example 5.1. The type of reduction described above has a familiar application, namely, the transformation of an evolutionary PDE in conservation form,

(5.2) \begin{equation} u_t=D_xF(x,t,[u]) \end{equation}

to its potential form. To see this, substitute $u=w_x$ into ( 5.2 ) to get

(5.3) \begin{equation} \mathcal{A}\,:\!=\,-w_{xt}+D_xF(x,t,[w_x])=0. \end{equation}

This PDE has a family of multipliers $\mathcal{Q}={\mathtt {g}}(t)$ , which yields the conservation law

(5.4) \begin{equation} D_x\{-w_{t}+F(x,t,[w_x])\}=0. \end{equation}

Thus, there is a first integral,

(5.5) \begin{equation} -w_{t}+F(x,t,[w_x])=f(t). \end{equation}

The PDE ( 5.3 ) also has the family of point symmetries $w\mapsto w+h(t)$ , which map ( 5.5 ) to

(5.6) \begin{equation} -w_{t}+F(x,t,[w_x])=f(t)+h'(t). \end{equation}

By choosing $h$ appropriately, it is clear that there is only one equivalence class, which contains the potential form of ( 5.2 ):

(5.7) \begin{equation} w_{t}=F(x,t,[w_x]). \end{equation}

So, the well-known transformation from ( 5.2 ) to ( 5.7 ) works because there are appropriate families of multipliers and symmetries.

Suppose that a member of an equivalence class is invariant under a Lie group of non-trivial point symmetries of (5.1) that depend on one or more arbitrary functions of one variable (whether or not these are symmetries of $\mathcal{A}=0$ ). Then, group splitting can be used to reduce (5.1) further. If the further-reduced PDE can be solved, all solutions in the relevant equivalence class can be reconstructed.

If $\mathcal{A}=0$ is an Euler–Lagrange equation, the multiplier depending on ${\mathtt {g}}(t)$ is also a variational symmetry characteristic (by Noether’s Theorem). Therefore, partitioning into a small number of equivalence classes is always possible. If an equivalence class is invariant under the symmetries that depend on ${\mathtt {g}}(t)$ , group splitting gives a true double reduction for that class which mirrors the double reduction of order for Euler–Lagrange ODEs.

6. Pseudoparabolic examples

We illustrate the above general principles through application to pseudoparabolic PDEs of the form

(6.1) \begin{equation} \mathcal{A}\,:\!=\, -u_t+ D_x\{M(u) D_xD_t\Psi (u) \}=0,\qquad M(u)\Psi '(u)\neq 0. \end{equation}

Such equations, typically with an additional diffusion term, arise in a variety of applications (see e.g. Barenblatt et al. [Reference Barenblatt, Bertsch, Passo, Prostokishin and Ughi5] and Cuesta and Hulshof [Reference Cuesta and Hulshof6]), with the limit case of negligible diffusivity, as in (6.1), being of specific interest (e.g. King [Reference King10]). The general theory furnishes results for these PDEs that we believe to be new.

We begin with the restriction

(6.2) \begin{equation} M(u)=\Psi '(u), \end{equation}

which enables (6.1) be written as an Euler–Lagrange equation by substituting $u=w_x$ , as follows:

(6.3) \begin{equation} \mathcal{A}\,:\!=\, -w_{xt}+ D_x\{\Psi '(w_x) D_xD_t\Psi (w_x) \}=0,\qquad \Psi '(w_x)\neq 0. \end{equation}

The Lagrangian functional is

(6.4) \begin{equation} \mathcal{L}=\iint \tfrac {1}{2}w_xw_t+ \tfrac {1}{2} \left (\Psi '(w_x)\right)^2 w_{xx}w_{xt}\, \mathrm{d} x\, \mathrm{d} t, \end{equation}

which is invariant under translations in $x$ . By Noether’s Theorem, the corresponding multiplier $\mathcal{Q}=w_x$ gives the conservation law

(6.5) \begin{equation} D_t\{-w_x^2-\tfrac {1}{2}(D_x\Psi (w_x))^2\}+D_x\{w_x\Psi '(w_x)D_xD_t\Psi (w_x)\}=0 \end{equation}

There are two families of multipliers that depend on arbitrary functions of $t$ . The first is described in Example 5.1 and yields the potential version of (6.1), namely

(6.6) \begin{equation} w_{t}= \Psi '(w_x) D_xD_t\Psi (w_x). \end{equation}

The second family of multipliers, $\mathcal{Q}={\mathtt {g}}(t)w_t$ , are characteristics for the variational symmetries

(6.7) \begin{equation} t\mapsto h(t),\qquad h'(t)\neq 0, \end{equation}

which are of course also symmetries of (6.1). The conservation law arising from (3.4) is

(6.8) \begin{equation} D_x\left \{{\mathtt {g}}(t)\left (-\tfrac {1}{2}w_t^2+w_t\Psi '(w_x)D_xD_t\Psi (w_x)-\tfrac {1}{2}(D_t\Psi (w_x))^2\right)\right \}=0, \end{equation}

which gives the first integral

(6.9) \begin{equation} -w_t^2+2w_t\Psi '(w_x)D_xD_t\Psi (w_x)-(D_t\Psi (w_x))^2=f(t). \end{equation}

Taking (6.6) into account simplifies this to

(6.10) \begin{equation} w_t^2-(D_t\Psi (w_x))^2=f(t). \end{equation}

The variational symmetries (6.7) map (6.10) to

(6.11) \begin{equation} w_t^2-(D_t\Psi (w_x))^2=f(t)(h'(t))^{-2}, \end{equation}

splitting the first integral into three equivalence classes. For simple representatives of each class, we choose $f(t)\in \{0, \pm 1\}$ in (6.10). The class with $f(t)=0$ consists of two cases:

(6.12) \begin{equation} D_t\Psi (w_x)\pm w_t=0, \end{equation}

each of which is consistent with (6.6) and admits the above variational symmetries. Consequently, this class reduces to a first-order ODE:

(6.13) \begin{equation} \Psi (w_x)\pm w=g(x), \end{equation}

where $g(x)$ is arbitrary.

Reinstating $u$ as the dependent variable, (6.10) with a non-zero right-hand side leads to two possibilities:

\begin{equation*} u_t=\pm D_x\!\left (\left \{\left (D_t\Psi \left (u\right)\right)^2 +f(t)\right \}^\frac {1}{2}\right),\qquad f(t)\neq 0. \end{equation*}

From (6.13), the remaining possibilities are

\begin{equation*} D_x\Psi (u)\pm u=g'(x). \end{equation*}

We now turn to the general case of (6.1), in which (6.2) need not hold. A standard calculation (as in Anco [Reference Anco, Melnik, Makarov and Belair1]) shows that, for arbitrary $M$ and $\Psi$ , all multipliers that depend on $(x,t,u)$ only are linear combinations of

(6.14) \begin{equation} \mathcal{Q}_1=1,\qquad \mathcal{Q}_2=\int \frac {\Psi '(u)}{M(u)}\,\mathrm{d} u. \end{equation}

The corresponding conservation laws are $\mathcal{A}=0$ and

(6.15) \begin{equation} D_t\{-\Phi _2(u)-\tfrac {1}{2}(D_x\Psi (u))^2\}+D_x\{\Phi _2'(u)M(u)D_xD_t\Psi (u)\}=0,\quad \text{where}\ \Phi _2(u)=\int \mathcal{Q}_2(u)\,\mathrm{d} u. \end{equation}

This conservation law amounts to (6.5) when the restriction (6.2) holds.

There are two special cases, each of which has a family of additional multipliers depending on an arbitrary function ${\mathtt {g}}(x)$ . If $M(u)=1$ , the family is $\mathcal{Q}={\mathtt {g}}(x)$ , so, (3.4) yields the (obvious) conservation law

(6.16) \begin{equation} D_t\{{\mathtt {g}}(x)(\!-u+D_x^2\Psi (u))\}=0. \end{equation}

The resulting first integral,

(6.17) \begin{equation} -u+D_x^2\Psi (u)=f(x), \end{equation}

reduces the PDE to a family of ODEs.

The second special case occurs when $M(u)=\exp (-\mu \Psi (u))$ , where $\mu$ is a non-zero constant. This gives the family of multipliers $\mathcal{Q}={\mathtt {g}}(x)\mu ^{-1}\exp (\mu \Psi (u))$ . The corresponding conservation laws (3.4) are

(6.18) \begin{equation} D_t\left \{{\mathtt {g}}(x)\left (\mu ^{-1}D_x^2\Psi (u)-\tfrac {1}{2}(D_x\Psi (u))^2- \Phi (u)\right)\right \}=0,\quad \text{where}\ \Phi (u)=\int \mu ^{-1}\exp (\mu \Psi (u))\,\mathrm{d} u. \end{equation}

This leads to the first integral

(6.19) \begin{equation} \mu ^{-1}D_x^2\Psi (u)-\tfrac {1}{2}(D_x\Psi (u))^2-\Phi (u)=f(x). \end{equation}

Again, a family of multipliers depending on ${\mathtt {g}}(x)$ has reduced the PDE to an ODE. Indeed, such a reduction always occurs when a PDE has $t$ -derivatives of at most first order and multipliers that depend on an arbitrary ${\mathtt {g}}(x)$ . Every PDE (6.1) has the family of point symmetries (6.7). However, these are trivial symmetries of the reduced ODEs (6.17) and (6.19), so, they do not provide any further simplification.

The doubly exceptional instances of these special cases, in the sense that (6.2) also applies, are $M(u)=1$ , which gives the linear PDE

\begin{equation*} u_t=u_{xxt}\,,\end{equation*}

and

\begin{equation*} \Psi =\ln (\mu u) /\mu , \qquad \Phi =u^2 /2, \end{equation*}

so that $M(u)=\exp (\!-\mu \Psi (u))=1 /(\mu u)$ . In the latter case, setting $u=1 / \sigma ^{2}$ in (6.19) leads to Pinney’s equation in the form

\begin{equation*} \sigma _{xx}+\frac {\mu ^{2} f(x)}{2}\,\sigma +\frac {\mu ^{2}}{4 \sigma ^{3}}=0, \end{equation*}

implying the integrability of the PDE. Special cases arise from (6.19) much more generally, in fact. Setting

\begin{equation*} \Psi (u)= -\frac {2}{\mu }\ln \sigma \end{equation*}

gives

\begin{equation*} \sigma _{xx}+\frac {\mu ^{2}f(x)}{2}\,\sigma +\frac {\mu ^{2}}{2}\,\sigma F (\sigma)=0 \end{equation*}

where $F(\sigma)=\Phi (u)$ , so a second integrable case arises for $F(\sigma)=1 / \sigma$ , whereby, setting $\mu =-1$ ,

\begin{equation*} u_t= 2 D_x\!\left (u^{2}D_xD_t\ln u\right) \end{equation*}

leads via $\sigma =u$ to

\begin{equation*} u_{xx}+\frac {f (x)}{2}\,u=-\,\frac {1}{2}\,. \end{equation*}

That (6.7) represents a symmetry of (6.1) and (6.6) for any non-constant $h(t)$ allows a reduction of order in general, as in the familiar ODE theory. Since $x, u, w$ and $p\,:\!=\, u_x$ are each invariant under (6.7), the upshot is that the third-order PDE (6.1) can be reduced to a second-order one (in divergence form) for $p(u,x)$ , namely

(6.20) \begin{equation} D_x\!\left \{ M(u)D_u\left (p\Psi '(u) \right)\right \}+ D_u\!\left \{p M(u)D_u\! \left (p\Psi '(u) \right)\right \}=1, \end{equation}

while for (6.6) it follows that $u(w,x)$ satisfies

(6.21) \begin{equation} \Psi '(u)D_w\!\left \{D_x \Psi (u)+uD_w\Psi (u) \right \}=1. \end{equation}

That these two expressions are equivalent when $M(u)=\Psi '(u)$ follows on defining $\phi (u,x)$ by

(6.22) \begin{equation} \phi _u=\Psi '(u)\,D_u\!\left (p\Psi '(u)\right)\!, \end{equation}

whereby integration of (6.20) with respect to $u$ gives

(6.23) \begin{equation} \phi _x + p\,\phi _u=u, \end{equation}

without loss of generality. The left-hand side of (6.23) is $\phi _x$ at fixed $t$ ; so, we can set $\phi =w$ and (6.21) and (6.22) are then equivalent. The above reductions proceed by eliminating $t$ , the dependence upon which is reinstated as the function of integration resulting on solving

\begin{equation*} u_x=p(u,x) \quad \text{or}\quad w_x=u(w,x), \end{equation*}

thereby reconstituting the third-order nature of the original PDE.

We conclude here with some comments about travelling-wave solutions

\begin{equation*} u=u(z), \quad z=x-s(t), \end{equation*}

for which the corresponding special cases of the above results can be obtained from elementary ODE considerations: (6.1) becomes

(6.24) \begin{equation} \frac {du}{dz}=\frac {d}{dz}\left (M(u)\frac {d^{2}}{dz^{2}}\Psi (u)\right) \end{equation}

so that

(6.25) \begin{equation} 1=\frac {d}{du}\left (p\, M (u) \frac {d}{du}\left (p\,\Psi '(u) \right)\right), \end{equation}

the special case of (6.20) that arises when $p=p (u)$ . The obvious first integrals of (6.24) and (6.25) are also equivalent, and when (6.2) holds a further integration in the form

\begin{equation*} u^2 + \alpha u + \beta = \left (p\,\Psi '(u) \right)^{2}, \end{equation*}

for constant $\alpha$ and $\beta$ is also immediate, corresponding to the appropriate special case of (6.10). As $p=\mathrm{d} u/\mathrm{d} z$ , this case reduces completely to quadrature:

\begin{equation*} x-s(t)=c\pm \int \frac {\Psi (u)}{\sqrt {u^2+\alpha u+\beta }}\,\mathrm{d} u. \end{equation*}

7. Further examples

Example 7.1. Every generalised Liouville equation in the hierarchy

(7.1) \begin{equation} \mathcal{A}^{(k)}\,:\!=\,D_x^{2k}u_{xy}-e^u=0,\qquad k\geq 0, \end{equation}

has a variational formulation, with the Lagrangian

\begin{equation*} L=\tfrac {1}{2}(D_x^{2k}u)u_{xy}-e^u. \end{equation*}

Such equations admit the variational symmetries

(7.2) \begin{equation} \hat {x}=x,\qquad \hat {y}=h(y) \qquad \hat {u}=u-\ln \left (h^{'}(y)\right),\qquad h'(y)\neq 0, \end{equation}

and the corresponding multiplier is $\mathcal{Q}=g'(y)+g(y)u_y$ . Applying Theorem 3.1 gives the first integral (for $k\geq 1$ )

(7.3) \begin{equation} \lambda = -D_x^{2k}u_{yy}+u_yD_x^{2k}u_y-\cdots +(\!-1)^{k-1}(D_x^{k-1}u_y)(D_x^{k+1}u_y)+\tfrac {1}{2}(\!-1)^k(D_x^k u_y)^2=f(y). \end{equation}

For $k=0$ , the first integral is

(7.4) \begin{equation} \lambda =-u_{yy}+\tfrac {1}{2}u_y^2=f(y). \end{equation}

For each $k$ , the variational symmetries ( 7.2 ) map $\lambda$ to $(h'(y))^{-2}\lambda$ , so take $f(y)\in \{0,\pm 1\}$ as representatives of each equivalence class. In particular, when $f(y)=0$ , these symmetries give a reduction of order (with respect to $x$ ) by group splitting. To see this in action, we examine what happens for $k=0$ and $k=1$ .

The lowest-order invariants of the symmetries are

\begin{equation*} x,\qquad p=u_x,\qquad q=u_{xy}e^{-u}\qquad r=u_{xx}. \end{equation*}

We derive a system of reduced equations for $q(x,p)$ and $r(x,p)$ . The relationship between $q$ and $r$ is expressed by the first-order integrability condition

(7.5) \begin{equation} qr_p=u_{xxy}e^{-u}=q_x+rq_p+pq. \end{equation}

The case $k=0$ (the Liouville equation) amounts to $q=1$ ; so, the integrability condition has the solution

\begin{equation*} r=\tfrac {1}{2}p^2+F(x), \end{equation*}

where $F$ is arbitrary. Together with the first integral ( 7.4 ), this gives the well-known reduction of the Liouville equation to a pair of ODEs.

More interestingly, the symmetry reduction for the case $k=1$ gives the ODE

(7.6) \begin{equation} qr_{pp}+q_pr_p+\tfrac {1}{2}q=0, \end{equation}

coupled with the integrability condition ( 7.5 ). This is supplemented by the first integral ( 7.3 ), namely

\begin{equation*} \lambda =-u_{xxyy}+u_yu_{xxy}-\tfrac {1}{2}u_{xy}^2=f(y). \end{equation*}

Example 7.2. Consider the following time-independent curvature equation, which is relevant to capillary surfaces, for example (see King et al. [12]):

(7.7) \begin{equation} \mathcal{A}\,:\!=\,\kappa (x,y)-\nabla \cdot \left (\frac {\nabla u}{|\nabla u|}\right)=0. \end{equation}

This expresses the curvature of plane curves of constant $u$ (i.e. level sets), in terms of a given function $\kappa (x,y)$ . Equation ( 7.7 ) is the Euler–Lagrange equation corresponding to the functional

(7.8) \begin{equation} \mathcal{L}=\iint \left (|\nabla u|+\kappa (x,y)u\right)\,\mathrm{d} x\,\mathrm{d} y. \end{equation}

While ( 7.7 ) is evidently invariant under $u\rightarrow U(u),\ U'(u)\neq 0$ , these symmetries are not variational.

In the special case where $\kappa$ is independent of $x$ , there exists a family of multipliers, $\mathcal{Q}={\mathtt {g}}(u)u_x$ . These are non-trivial in any neighbourhood where $u_x\neq 0$ , so a conservation law-reduction can be obtained by a hodograph transformation, treating $u$ as an independent variable and $x$ as the dependent variable. To transform ( 7.7 ), without restricting $\kappa$ in advance, let $\phi (u,y)=\kappa (x(u,y),y)$ to obtain the following family of conservation laws that hold, remarkably, for all $\kappa (x,y)$ :

(7.9) \begin{equation} D_y\left \{{\mathtt {g}}(u)\left (\frac {x_y}{(1+x_y^2)^{1/2}}+\Phi (u,y)\right)\right \}=0,\quad \text{where}\quad \Phi (u,y)=\int \phi (u,y)\,\mathrm{d} y. \end{equation}

This leads to the first integral

\begin{equation*} \frac {x_y}{(1+x_y^2)^{1/2}}+\Phi (u,y)=f_1(u), \end{equation*}

from which ( 7.7 ) can be solved by quadrature:

\begin{equation*} x=f_2(u)\pm \int \frac {f_1(u)-\Phi (u,y)}{\{1-(f_1(u)-\Phi (u,y))^2\}^{1/2}}\,\mathrm{d} y. \end{equation*}

Here, $f_1$ and $f_2$ are arbitrary, subject to the constraint $|f_1(u)-\Phi (u,y)|\lt 1$ (which amounts to $u_x$ being real and non-zero). A similar reduction, with $y=y(x,u)$ , is applicable in neighbourhoods where $u_y\neq 0$ .