Remark 2.2. We may also write

(2.3) $$ \begin{align} S(\rho) = \left(z^{-1}\rho_{xx}\rho_x^{-1}z\right)_x -\frac{1}{2}\left(z^{-1}\rho_{xx}\rho_x^{-1}z\right)^2, \qquad{} \mathrm{where} \qquad{} z_x=\frac{1}{2}\rho_{xx}\rho_x^{-1}z, \end{align} $$

or

(2.4) $$ \begin{align} S(\rho) = \left(y\rho_x^{-1}\rho_{xx}y^{-1}\right)_x -\frac{1}{2}\left(y\rho_x^{-1}\rho_{xx}y^{-1}\right)^2, \qquad{} \mathrm{where} \qquad{} y_x=\frac{1}{2}y\rho_x^{-1}\rho_{xx}, \end{align} $$

formulations obtained by, respectively, substituting for $y=z^{-1}\rho _x$ and $z=\rho _xy^{-1}$ in (2.1) and $z_x y = z y_x$ .

Remark 2.3. The formulation (2.1) is simpler than the Lagrange Schwarzian derivative introduced in [Reference Ovsienko20] with fewer assumptions being made on $\rho $ . In [Reference Ovsienko20], $\rho $ is assumed to be symmetric and such that $\rho _x$ is positive definite, although this last condition can be relaxed [Reference Marí Beffa17]. For our definition of matrix Schwarzian derivative, it is sufficient to assume that $\rho $ is such that the linear equation $z_x=\frac {1}{2}\rho _{xx}\rho _x^{-1}z$ admits a fundamental matrix solution z in some domain (see, e.g., [Reference Hille11]). We may then take $y=z^{-1}\rho _x$ . We assume $\rho $ to be analytic with $\rho _x$ nonsingular in some domain D, so that $\frac {1}{2}\rho _{xx}\rho _x^{-1}$ is also analytic in D. A solution z, locally analytic and nonsingular in D, can then be constructed. This solution may be multi-valued but, if necessary, we may restrict our attention to some subdomain of D in order to obtain a single-valued solution. It is worth noting that, even if z is multi-valued, the quantity $z^{-1}z_x$ might be single-valued, and so also the matrix Schwarzian derivative, since this can be rewritten in the form $S(\rho )=2(z^{-1}z_x)_x-2(z^{-1}z_x)^2$ (and also, indeed, $S(\rho )=2(y_xy^{-1})_x-2(y_xy^{-1})^2$ ). We note that Definition 2.1 also provides an alternative definition of matrix Schwarzian derivative to that presented in [Reference Carillo, Lo Schiavo and Schiebold4].

Remark 2.5. The above result corresponds to the fact that, given any solution s of $s_x=\frac {1}{2}\rho _{xx}\rho _x^{-1}s$ , then $z=sP$ gives a solution of $z_x=\frac {1}{2}\rho _{xx}\rho _x^{-1}z$ . Since s and z are assumed nonsingular, then so is P. Then $zy=st$ gives $t=Py$ , and so $T(\rho )=s^{-1}R(\rho )t^{-1}=PS(\rho )P^{-1}$ .

Remark 2.6. Proposition 2.4 is the analog of the result in [Reference Ovsienko20] that the (more restrictive) Lagrange Schwarzian derivative is defined up to conjugation by orthogonal matrices.

Remark 2.11. From (2.8) and (2.9), we see that the condition that the matrix $(AC^{-1}D-B)$ be nonsingular means that the transformations $M_1$ and $M_2$ are non-constant, since $(AC^{-1}D-B)$ cannot be zero. (The matrix $(B-AC^{-1}D)$ has an interpretation in the theory of quasideterminants, see, e.g., the discussion in [Reference Gelfand, Gelfand, Retakh and Wilson6].)

Remark 2.12. Proposition 2.10, while very similar to Theorem 4.2 in [Reference Carillo, Lo Schiavo and Schiebold4], is in fact different from it. In Proposition 2.10, we show the invariance of the matrix Schwarzian derivative itself; in [Reference Carillo, Lo Schiavo and Schiebold4], the authors’ interest is in the invariance of their non-Abelian KdV singularity manifold equation, and their Schwarzian derivative is not itself invariant. (The same remark holds with respect to Lemmas 2.7–2.9 when compared to Propositions 4.4 and 4.5 in [Reference Carillo, Lo Schiavo and Schiebold4].) These results are consistent since Definition 2.1 does not include the definition used in [Reference Carillo, Lo Schiavo and Schiebold4], as the condition $z_xy=zy_x$ excludes the choice $z=\rho _x$ and $y=I$ (except when $\rho _{xx}=0$ and so $S(\rho )=0$ ). For the Lagrange Schwarzian derivative, the result corresponding to Proposition 2.10 is Proposition 1, part i), in [Reference Ovsienko20].

Remark 2.14. Proposition 2.13 corresponds to Proposition 1, part ii), in [Reference Ovsienko20] for the Lagrange Schwarzian derivative.

Remark 2.16. Proposition 2.15 has no counterpart in [Reference Ovsienko20], as the Lagrange Schwarzian derivative is defined for symmetric matrices, and so is in fact invariant under transposition of $\rho $ . As for the matrix Schwarzian derivative defined in [Reference Carillo, Lo Schiavo and Schiebold4], similarly to Proposition 2.10 (see Remark 2.12), this is not invariant, nor gives the transpose, under $\rho =r^T$ , whereas the non-Abelian KdV singularity manifold equation, expressed in terms of this matrix Schwarzian derivative, is.

Remark 3.4. The last term in (3.16) cancels with $g_{n-1}I$ arising from $(\partial _x+A_u)g_{n-1}xI=2g_{n-1}xu+g_{n-1}I$ . Thus, (3.16) may also be written (here each $M_k$ is evaluated at $w=M[u]$ ) as

(3.17) $$ \begin{align} (\partial_x+A_u)\left(M_n+\sum_{k=0}^{n-1} c_k M_k\right)+uE+Eu +2g_{n-1}xu-z^{-1}C_1 z -y C_2y^{-1}=0. \end{align} $$

It is the term $2g_{n-1}xu$ that motivates the requirement $g_{n-1}\neq 0$ when defining (3.16) to be a matrix second Painlevé hierarchy: this nonautonomous term is fundamental to scalar and matrix higher-order analogs of the second Painlevé equation of the form considered here, and, of course, to the second Painlevé equation itself. If $g_{n-1}\neq 0$ (all previously-known examples of) such equations have a Lax pair of the form $\psi _x=F\psi $ , $\psi _\lambda =G\psi $ .

Remark 3.5. We justify Definition 3.3, that is, of (3.16) as a matrix second Painlevé hierarchy (when $g_{n-1}\neq 0$ ), in Remark 3.12 below. As we now see in Proposition 3.6, the relationship between our matrix second Painlevé hierarchy and its Schwarzian counterpart (Definition 3.2) mirrors that of the scalar case (see [Reference Kudryashov and Pickering13], and also [Reference Weiss25] for the case of the second Painlevé equation itself), that is, solutions of the latter give rise to solutions of the former. The above Schwarzian matrix second Painlevé hierarchy is completely new. The above matrix second Painlevé hierarchy is also new, due to its greatly generalized parameter dependence—via the arbitrary matrices $C_1$ and $C_2$ —when compared to the matrix second Painlevé hierarchy in [Reference Gordoa, Pickering and Zhu9], again as explained in Remark 3.12.

Remark 3.8. The stationary (time-independent) reduction of the hierarchy (3.24) gives the hierarchy (3.15).

Remark 3.11. The hierarchy (3.31) is a generalization of the matrix mKdV hierarchy (3.12) (see (3.14)), being nonlocal (when $E\neq 0$ ) and nonisospectral (when $g_{n-1}\neq 0$ ). Acting with $M'[u]=(\partial _x-A_u)$ on (3.31) yields the result used in [Reference Gordoa, Pickering and Zhu9] that, just as in Definition 3.1 for the matrix mKdV and KdV hierarchies, the Miura map $w=M[u]=u_x-u^2$ maps from solutions of (3.31) to solutions of the nonlocal nonisospectral matrix KdV hierarchy $w_t=\mathcal {B}_1[w]K[w]$ (for the local non-isospectral matrix KdV hierarchy obtained by setting $E=0,$ we refer to [Reference Jingping, Chou and Youjin12]). The special case $E=0$ , $g_{n-1}=0,$ and $C_1=C_2=C_3=0$ of Proposition 3.10 corresponds to a result given in [Reference Marí Beffa17], but where the Lagrange Schwarzian derivative is used; a similar result can be found in [Reference Carillo, Lo Schiavo and Schiebold4].

Remark 3.12. It was suggested in [Reference Levi, Ragnisco and Rodríguez16] (see also [Reference Gordoa and Pickering7]) that the stationary flows of nonisospectral integrable PDEs could be used to derive Painlevé equations. It was thus that in [Reference Gordoa, Pickering and Zhu9] we considered the stationary (time-independent) reduction of (3.31), that is, $\mathcal {B}[u](M'[u])^\dagger K[M[u]]=0$ , or

(3.35) $$ \begin{align} (\partial_x-C_u\partial_x^{-1}C_u)(\partial_x+A_u)K[M[u]]=0, \end{align} $$

wherein in $K[w]$ , given by (3.14), E and $g_{n-1}$ are now assumed constant. By observing that $(g_{n-1}+\alpha _n)I\in \mathrm {Ker} [\mathcal {B}[u]]$ , we were led to the hierarchy

(3.36) $$ \begin{align} (\partial_x+A_u)K[M[u]]-(g_{n-1}+\alpha_n)I=0, \end{align} $$

which we defined, when $g_{n-1}\neq 0$ , to be a matrix second Painlevé hierarchy. This result, as explained in [Reference Gordoa, Pickering and Zhu9], corresponds to identifying part (but not all) of $\mathrm {Ker} [\mathcal {B}[u]]$ . In Proposition 3.13 below, we show that $\mathrm {Ker} [\mathcal {B}[u]]$ may be written $z^{-1}K_1 z +y K_2y^{-1}$ , where $K_1$ and $K_2$ are arbitrary square matrices. Equation (3.35) thus yields

(3.37) $$ \begin{align} (\partial_x+A_u)K[M[u]]-z^{-1}K_1 z -y K_2y^{-1}=0, \end{align} $$

with equation (3.16) corresponding to setting, for example, $K_1=C_1+g_{n-1}I$ and $K_2=C_2$ ; equation (3.36) may then be obtained by taking $C_1=\alpha I$ and $C_2=\beta I$ , $\alpha $ and $\beta $ scalar constants, and then setting $\alpha +\beta =\alpha _n$ . Proposition 3.13 is stronger than the reverse process, that is, the steps used above in the proof of Proposition 3.10, and allows us to claim that equation (3.16), for $g_{n-1}\neq 0$ , defines a new, more general matrix second Painlevé hierarchy (when compared with (3.36)). We remark that the Schwarzian equation corresponding to (3.36) is given by $zK[\frac {1}{2}S(\rho )]y-(\alpha _n+g_{n-1})\rho -C_3=0$ (if $\alpha _n+g_{n-1}\neq 0,$ we may set $C_3=0$ using a shift on $\rho $ ; see Lemma 2.9).

Remark 3.14. The procedure introduced in [Reference Pickering22] can be used to invert $(\partial _x+A_u)$ and so obtain (3.15) from (3.16). This procedure is based on the observation that if K, $L,$ and M depend on x and u and its derivatives, and if s and t satisfy $s_x=sM$ and $t_x=Nt$ , then $(sKt)_x=s\left [(\partial _x+L_M+R_N)K\right ]t$ . The first equality in (3.40) then corresponds to $K=\widetilde L$ , $M=u,$ and $N=-u$ where, for z and y satisfying $z_x=zu$ and $y_x=uy$ , we have taken $s=z$ and $t=z^{-1}$ so $s_x=su$ and $t_x=-z^{-1}z_xz^{-1}=-ut$ . Similarly, the first equality in (3.42) corresponds to $K=L$ , $M=-u,$ and $N=u$ , where we have taken $s=y^{-1}$ and $t=y$ so $s_x=-y^{-1}y_xy^{-1}=-su$ and $t_x=ut$ .

Remark 3.17. When $E=0,$ the Schwarzian matrix second Painlevé hierarchy and the Schwarzian matrix KdV hierarchy, that is,

(3.47) $$ \begin{align} zK\left[\frac{1}{2}S(\rho)\right]y-C_1\rho -\rho C_2 -g_{n-1}\rho-C_3=0, \end{align} $$

and

(3.48) $$ \begin{align} \frac{1}{2}\rho_t=zK\left[\frac{1}{2}S(\rho)\right]y-C_1\rho -\rho C_2 -g_{n-1}\rho-C_3, \end{align} $$

respectively, do not depend on z and y. In order to see this, we note that for any matrix function A,

(3.49) $$ \begin{align} (z^{-1}Ay^{-1})_x = -z^{-1}z_xz^{-1}Ay^{-1}+z^{-1}A_xy^{-1}- z^{-1}Ay^{-1}y_xy^{-1} = z^{-1}\left(-\frac{1}{2}\rho_{xx}\rho_x^{-1}A+A_x- \frac{1}{2}A\rho_x^{-1}\rho_{xx}\right)y^{-1}. \end{align} $$

Since $S(\rho )$ has the form $z^{-1}Ay^{-1}$ then so do all its derivatives, and it then follows that all terms resulting from the substitution $w=\frac {1}{2}S(\rho )$ in any $M_k$ , $k\geq 1$ , in $K[w]$ are products of quantities of the form $z^{-1}Ay^{-1}$ . In any such product, we may replace the combination $y^{-1}z^{-1}$ , which arises between adjacent quantities, by $\rho _x^{-1}$ . Thus, the result is a sum of terms which only depend on x-derivatives of $\rho $ , and having a left factor $z^{-1}$ and a right factor $y^{-1}$ , which are then canceled when constructing the combination $zK\left [\frac {1}{2}S(\rho )\right ]y$ in equations (3.47) and (3.48). Finally, the terms $M_0[w]+g_{n-1}xI$ in $K[w]$ gives rise to terms $\frac {1}{2}c_0\rho _x+g_{n-1}x\rho _x$ in these equations. A consequence of the above equations not depending on z and y is that when applying Propositions 3.6 and 3.10, we are free to choose z and y. Any alternative choice of s and t just yields an alternative variable $v=\frac {1}{2}s^{-1}\rho _{xx}t^{-1}$ for the matrix second Painlevé hierarchy or extended matrix mKdV hierarchy, respectively, this variable being related to u in Propositions 3.6 and 3.10 via $v=PuP^{-1}$ , where P is the constant matrix given in Proposition 2.4 (and the hierarchies satisfy $\mathcal {Q}[v,s,t,0,C_1,C_2]=P\mathcal {Q}[u,z,y,0,C_1,C_2]P^{-1}$ and $\mathcal {M}[v,0]=P\mathcal {M}[u,0]P^{-1}$ ).

Remark 3.18. As simple examples, let us present the first members of the hierarchies (3.15), (3.16), and (3.24). Our new Schwarzian matrix second Painlevé equation is

(3.50) $$ \begin{align} \frac{1}{2}\left(\rho_{xxx}-\frac{3}{2}\rho_{xx}\rho_x^{-1}\rho_{xx}\right)+ \frac{1}{2}c_0\rho_x+zEy+g_{n-1}x\rho_x-C_1\rho -\rho C_2-g_{n-1}\rho-C_3=0, \end{align} $$

where $zy=\rho _x$ and $z_xy=zy_x$ (note the $zEy$ term). Our corresponding new matrix second Painlevé equation is

(3.51) $$ \begin{align} u_{xx}-2u^3+c_0u+uE+Eu+2g_{n-1}xu-z^{-1}C_1 z -y C_2y^{-1}=0, \end{align} $$

where $z_x=zu$ and $y_x=uy$ . The first of these is more general than the Schwarzian matrix second Painlevé equation given in [Reference Pickering22], and the second is more general than the matrix second Painlevé equation obtained in [Reference Gordoa, Pickering and Zhu9], which in turn includes, as a further special case, the original version presented in [Reference Balandin and Sokolov2], [Reference Olver and Sokolov18]. As explained in Remark 3.12, these previous results may be recovered by taking, for example, $C_1=\alpha I$ and $C_2=\beta I$ , where $\alpha $ and $\beta $ are two scalar constants satisfying $\alpha +\beta =\alpha _n$ . Finally, our new Schwarzian matrix KdV equation is given by

(3.52) $$ \begin{align} \frac{1}{2}\rho_t= \frac{1}{2}\left(\rho_{xxx}-\frac{3}{2}\rho_{xx}\rho_x^{-1}\rho_{xx}\right)+ \frac{1}{2}c_0\rho_x+zEy+g_{n-1}x\rho_x-C_1\rho -\rho C_2-g_{n-1}\rho-C_3, \end{align} $$

where $zy=\rho _x$ and $z_xy=zy_x$ (again, note the $zEy$ term). This equation is a strong generalization of the Lagrangian Schwarzian KdV equation presented in [Reference Marí Beffa17], defined via the Lagrange Schwarzian derivative introduced in [Reference Ovsienko20], and also of the (equivalent, but differently-defined) non-Abelian KdV singularity manifold equation given in [Reference Carillo, Lo Schiavo and Schiebold4]. The equations in [Reference Carillo, Lo Schiavo and Schiebold4], [Reference Marí Beffa17] may be recovered, for example, by setting $E=0$ , $g_{n-1}=0$ , $C_1=C_2=C_3=0,$ and $c_0=0$ .