GBDT and explicit solutions for the matrix coupled dispersionless equations (local and nonlocal cases)

TL;DR

This paper introduces matrix coupled dispersionless equations, constructs explicit solutions, and analyzes their asymptotic behavior, covering local, nonlocal, and scalar cases to advance understanding of these integrable systems.

Contribution

It provides the first comprehensive explicit solutions and asymptotic analysis for matrix coupled dispersionless equations, including local and nonlocal variants.

Findings

Constructed explicit multipole solutions.

Derived explicit Darboux and wave matrix functions.

Analyzed asymptotic behavior in key cases.

Abstract

We introduce matrix coupled (local and nonlocal) dispersionless equations, construct wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and consider their asymptotics in some interesting cases. We consider the scalar cases of coupled, complex coupled and nonlocal dispersionless equations as well.

Equations

$$\displaystyle R_{x}=\frac{(-1)^{p}}{2}(VV_{t}+V_{t}V),\quad V_{tx}=\frac{1}{2}(VR+RV)\qquad\Big{(}R_{x}=\frac{\partial}{\partial x}R\Big{)},$$

$$R(x,t)=\mathrm{diag}\{{\rho }_1(x,t),{\rho }_2(x,t)\},V(x,t)=\left[\begin{array}{cc}0& v_1(x,t)\\ v_2(x,t)& 0\end{array}\right],$$

$$G_t-F_x+[G,F]=0,[G,F]:=GF-FG$$

$$w_x=G(x,t,\lambda )w;G(x,t,\lambda ):=\frac{\mathrm{i}}{4}\lambda j-\frac{\mathrm{i}}{2}{j}^{p+1}V(x,t),j:=\left[\begin{array}{cc}{I}_{m_1}& 0\\ 0& -I_{m_2}\end{array}\right]$$

$$\displaystyle w_{t}=F(x,t,\lambda)w,\quad F(x,t,\lambda):=\big{(}-\mathrm{i}jR(x,t)+j^{p}V_{t}(x,t)\big{)}\big{/}\lambda.$$

$$\displaystyle\rho_{x}+\frac{1}{2}\varkappa\big{(}|v|^{2}\big{)}_{t}=0,\quad v_{tx}=\rho v\quad(\rho=\overline{\rho},\quad\varkappa=\pm 1)$$

$$m_1=m_2=1,{\rho }_1={\rho }_2=\rho =\overline{\rho },v_1=v,v_2=\overline{v},p=(1+\varkappa )/2.$$

$$R(-x)=-R(x{)}^{\ast },v_1(x)=v(x),v_2(x)=v(-x{)}^{\ast }.$$

$$A_{1}S(0,0)-S(0,0)A_2={\Pi }_1(0,0){\Pi }_2(0,0{)}^{\ast }.$$

$$G=-\lambda q_1-q_0,F=-\frac{1}{\lambda }{Q}_{-1}.$$

$$q_1=-\frac{\mathrm{i}}{4}j,q_0=\frac{\mathrm{i}}{2}{j}^{p+1}V,Q_{-1}=\mathrm{i}jR-j^p{V}_t.$$

$$({\Pi }_1{)}_x=\sum _{i=0}^1{A}_1^i{\Pi }_1{q}_i,({\Pi }_1{)}_t=A_1^{-1}{\Pi }_1{Q}_{-1};$$

$$({\Pi }_2{)}_x=-\sum _{i=0}^1(A_2^{\ast }{)}^i{\Pi }_2{q}_i^{\ast },({\Pi }_2{)}_t=-(A_2^{\ast }{)}^{-1}{\Pi }_2{Q}_{-1}^{\ast };$$

$$S_x={\Pi }_1{q}_1{\Pi }_2^{\ast },S_t=-A_1^{-1}{\Pi }_1{Q}_{-1}{\Pi }_2^{\ast }{A}_2^{-1}.$$

$$D=\{(x,t):-\infty \le a_1<x<a_2\le \infty ,-\infty \le b_1<t<b_2\le \infty \},$$

$$A_{1}S(x,t)-S(x,t)A_2={\Pi }_1(x,t){\Pi }_2(x,t{)}^{\ast }$$

$$w_A(x,t,\lambda )=I_m-{\Pi }_2(x,t{)}^{\ast }S(x,t{)}^{-1}(A_1-\lambda I_n{)}^{-1}{\Pi }_1(x,t).$$

$$\frac{\partial }{\partial x}{w}_A(x,t,z)=\tilde{G}(x,t,z)w_A(x,t,z)-w_A(x,t,z)G(x,t,z),$$

$$\frac{\partial }{\partial t}{w}_A(x,t,z)=\tilde{F}(x,t,z)w_A(x,t,z)-w_A(x,t,z)F(x,t,z),$$

$$\tilde{G}=-\lambda {\tilde{q}}_1-{\tilde{q}}_0,\tilde{F}=-\frac{1}{\lambda }{\tilde{Q}}_{-1}.$$

$${\tilde{q}}_1=q_1=-\frac{\mathrm{i}}{4}j,{\tilde{q}}_0=q_0-(q_1{X}_0-X_0{q}_1),$$

$${\tilde{Q}}_{-1}=(I_m-X_{-1})Q_{-1}(I_m+Y_{-1}),$$

$$X_0:={\Pi }_2^{\ast }{S}^{-1}{\Pi }_1,X_{-1}:={\Pi }_2^{\ast }{S}^{-1}{A}_1^{-1}{\Pi }_1,Y_{-1}:={\Pi }_2^{\ast }{A}_2^{-1}{S}^{-1}{\Pi }_1.$$

$$(I_m-X_{-1})(I_m+Y_{-1})=I_m.$$

$$({\Pi }_2^{\ast }{S}^{-1}{)}_x=-{\tilde{q}}_1{\Pi }_2^{\ast }{S}^{-1}{A}_1-{\tilde{q}}_0{\Pi }_2^{\ast }{S}^{-1},({\Pi }_2^{\ast }{S}^{-1}{)}_t=-{\tilde{Q}}_{-1}{\Pi }_2^{\ast }{S}^{-1}{A}_1^{-1}.$$

$${\tilde{q}}_0=\frac{\mathrm{i}}{2}{j}^{p+1}\tilde{V},$$

$$\widetilde{V}=\left[\begin{array}[]{lr}0&\widetilde{v}_{1}\\ \widetilde{v}_{2}&0\end{array}\right]:=V+\frac{1}{2}j^{p}(X_{0}-jX_{0}j),\quad(X_{0}=\Pi_{2}^{*}S^{-1}\Pi_{1}).$$

$${\tilde{Q}}_{-1}=\mathrm{i}j\tilde{R}-j^p{\tilde{V}}_t.$$

$$\tilde{R}=\frac{1}{2\mathrm{i}}({\tilde{Q}}_{-1}j+j{\tilde{Q}}_{-1}),{\tilde{Q}}_{-1}=(I_m-X_{-1})Q_{-1}(I_m+Y_{-1}).$$

$$\frac{1}{2}({\tilde{Q}}_{-1}-j{\tilde{Q}}_{-1}j)=-j^p{\tilde{V}}_t.$$

$$({\Pi }_2^{\ast }{S}^{-1}{\Pi }_1{)}_t$$

$$=X_{-1}{Q}_{-1}-{\tilde{Q}}_{-1}{X}_{-1}.$$

$$({\Pi }_2^{\ast }{S}^{-1}{\Pi }_1{)}_t$$

$$=X_{-1}{Q}_{-1}+(I_m-X_{-1})Q_{-1}(I_m+Y_{-1})(I_m-X_{-1})-{\tilde{Q}}_{-1}.$$

Full text

MSC(2010): 35B06, 37K40

Keywords: matrix coupled dispersionless equation, matrix nonlocal dispersionless equation, complex dispersionless equation, Darboux matrix, transfer matrix function, wave fuction, exlicit solution, asymptotics.

1 Introduction

The coupled dispersionless equations (real and complex) are integrable systems, which are actively studied since the important works [15, 17] (see, e.g., [2, 4, 7, 14, 16, 19] and various references therein). These equations are of independent interest and play also an essential role in the study of the short pulse equations (see [15, 7] and the references therein). We consider first the matrix generalization of the coupled dispersionless equations (MCDE):

[TABLE]

where diag stands for the block diagonal matrix, the blocks $\rho_{k}$ are $m_{k}\times m_{k}$ matrix functions $(k=1,2;\,\,m_{k}>0)$, $v_{1}$ is an $m_{1}\times m_{2}$ matrix function, $v_{2}$ is an $m_{2}\times m_{1}$ matrix function, and $R$ and $V$ are $m\times m$ matrix functions $(m:=m_{1}+m_{2})$. Clearly, it suffices for $p$ in (1.1) to take one of the two values $\{0,1\}$, that is, $p$ is either [math] or $1$.

It is easy to see that system (1.1) is equivalent to the compatibility condition

[TABLE]

of the following auxiliary linear systems:

[TABLE]

(where $I_{k}$ is the $k\times k$ identity matrix), and

[TABLE]

The complex coupled dispersionless equations

[TABLE]

appear, when we set in (1.1), (1.4), and (1.5):

[TABLE]

Here $\overline{\rho(x,t)}$ is the complex conjugate of $\rho(x,t)$. We note that the generalized coupled dispersionless system in [15, 16] is more general than MCDE (1.1). However, MCDE is more concrete and it is well known also (see, e.g., [3]) that the matrix and multicomponent generalizations are of interest in applications.

For the case of MCDE, we introduce the GBDT-version of the Bäcklund–Darboux transformation and construct wide classes of explicit solutions and corresponding explicit expressions for the Darboux and wave matrix functions $w$. Various versions of Bäcklund–Darboux transformations and related commutation methods are presented, for instance, in [5, 6, 11, 12, 18, 21, 22, 33] (see also the references therein). For generalized Bäcklund–Darboux (GBDT) approach see, for instance, [25, 26, 30]. The Darboux matrices for the generalized coupled dispersionless systems given in [16] were constructed in [14] by an iterative procedure and for a special case of diagonal generalized eigenvalues. GBDT allows to achieve an essential progress in this respect since neither the diagonal structure of the generalized eigenvalues nor iterative procedure are required there.

Nonlocal nonlinear integrable equations have been actively studied during the last years (see the important papers [2, 4, 9, 10, 13] and numerous references therein), starting from the article [1] on the nonlocal nonlinear Schrödinger equation. The nonlocal (scalar) dispersionless equations were considered in [2, 4]. Here we consider the nonlocal case $R(-x)=-R(x)^{*}$, $V(-x)=V(x)^{*}$, that is,

[TABLE]

We develop further the nonlocal results from [23], introduce GBDT for the nonlocal equations (1.1), (1.8) and construct the corresponding explicit solutions and wave functions. The explicit construction of the wave functions is new even for the local and nonlocal scalar dispersionless equations.

In Subsection 2.3 we consider also asymptotics of the Darboux matrix functions (Darboux matrices) and, correspondingly, of the wave matrix functions (wave functions) in the case of explicit solutions.

In the paper, ${\mathbb{N}}$ denotes the set of natural numbers, ${\mathbb{R}}$ denotes the real axis, ${\mathbb{C}}$ stands for the complex plane, and ${\mathbb{C}}_{+}$ (${\mathbb{C}}_{-}$) stands for the open upper (lower) half-plane. The spectrum of a square matrix $A$ is denoted by $\sigma(A)$. The notation diag means that the matrix is diagonal (or block diagonal).

2 GBDT for the matrix

coupled dispersionless equations

2.1 Preliminaries

GBDT, which we consider here, is a particular case of the GBDT introduced in [26, Theorem 1.1]. After fixing some $n\in{\mathbb{N}}$, each GBDT for MCDE (1.1) is determined by the initial system (1.1) itself and by five parameter matrices with complex-valued entries: three $n\times n$ invertible parameter matrices $A_{1}$, $A_{2}$ and $S(0,0)$ ($\det A_{i}\not=0$, $\,i=1,2;$ $\det S(0,0)\not=0)$, and two $n\times m$ parameter matrices $\Pi_{1}(0,0)$ and $\Pi_{2}(0,0)$ such that

[TABLE]

Similar to [26], we introduce coefficients $q_{1}(x,t)$, $q_{0}(x,t)$ and $Q_{-1}(x,t)$ via $G$ and $F$:

[TABLE]

Hence, in view of (1.4) and (1.5) we have

[TABLE]

If (1.1) holds (and $V_{t}$ is continuous with respect to both variables combined), then the following linear differential systems are compatible and (jointly with the initial values $S(0,0)$, $\Pi_{1}(0,0)$, and $\Pi_{2}(0,0)$) determine matrix functions $S(x,t)$, $\Pi_{1}(x,t)$, and $\Pi_{2}(x,t)$, respectively:

[TABLE]

Although the point $x=0$, $t=0$ is chosen above as the initial point, it is easy to see that any other point may be chosen for this purpose as well. Consider $S(x,t)$, $\Pi_{1}(x,t)$, and $\Pi_{2}(x,t)$ in some domain $D$, for instance,

[TABLE]

such that $R(x,t)$ and $V(x,t)$ of the form (1.2) are well defined in $D$ and satisfy (1.1), and such that $(0,0)\in D$. Then $S(x,t)$, $\Pi_{1}(x,t)$, and $\Pi_{2}(x,t)$ are well defined and the identity

[TABLE]

follows from (2.1) and (2.3)–(2.6) [26].

Introduce (in the points of invertibility of $S(x,t)$ in $D$) the matrix functions

[TABLE]

In view of (2.7), the matrix function $w_{A}(x,t,\lambda)$ is the so called transfer matrix function in Lev Sakhnovich’s form (see [30, 31, 32] and the references therein) at each point $(x,t)$ of invertibility of $S(x,t)$. Our next proposition refers to the particular case of [26, (1.34)].

Proposition 2.1**.**

Let $R$ and $V$ have the form (1.2) and satisfy the MCDE (1.1). Assume that $S(x,t)$, $\Pi_{1}(x,t)$ and $\Pi_{2}(x,t)$ satisfy (2.1) and (2.4)–(2.6), and that $w_{A}$ is given by (2.8). Then, in the points of invertibility of $S(x,t)$ we have

[TABLE]

where $G$ and $F$ are given by (2.2) and (2.3),

[TABLE]

$q_{0}$* and $Q_{-1}$ are defined by (2.3), and $X_{0}$, $X_{-1}$ and $Y_{-1}$ are given by the formulas*

[TABLE]

According to (2.2) and (2.11), $G$ and $\widetilde{G}$ and $F$ and $\widetilde{F}$ depend on $\lambda$ in a similar way (e.g., $G$ and $\widetilde{G}$ are polinomials of the first degree with respect to $\lambda$ and their leading coefficients coincide).

Compare (2.8) and (2.14) in order to see that $w_{A}(x,t,0)=I_{m}-X_{-1}(x,t)$. Hence, the equality [30, (1.76)] (at $z=\zeta=0$) yields

[TABLE]

Finally, according to [26, (1.17)] the following useful relations are valid:

[TABLE]

2.2 Darboux matrix

It follows from (2.2), (2.3) and (2.11), (2.12) that $\widetilde{q}_{0}$ has the same form as $q_{0}$, and so $\widetilde{G}$ has the same form as $G$. More precisely, in the expression for $\widetilde{q}_{0}$ we substitute only $\widetilde{V}$ instead of $V$, $\widetilde{v}_{1}$ instead of $v_{1}$ and $\widetilde{v}_{2}$ instead of $v_{2}$, that is

[TABLE]

where

[TABLE]

According to (2.2), (2.3) and (2.11), the proposition below means that $\widetilde{F}$ has the same form as $F$ and $\widetilde{Q}_{-1}$ has the same form as $Q_{-1}$.

Proposition 2.2**.**

Let the conditions of Proposition 2.1 hold. Then, in the points of invertibility of $S$, we have

[TABLE]

where $\widetilde{V}$ is given by (2.18) and

[TABLE]

Proof.

The second equality in (2.20) coincides with (2.13). In view of the first equality in (2.20) the block diagonal part of $\widetilde{Q}_{-1}$ equals $\mathrm{i}j\widetilde{R}$ and in order to prove (2.19) it remains to show that the block antidiagonal part of $\widetilde{Q}_{-1}$ equals $-j^{p}\widetilde{V}_{t}$, that is,

[TABLE]

First, let us find the derivative $(\Pi_{2}^{*}S^{-1}\Pi_{1})_{t}$. Second equalities in (2.4) and (2.16) and the definition of $X_{-1}$ in (2.14) yield

[TABLE]

Using the second equality in (2.20) and formula (2.22) we derive

[TABLE]

By virtue of (2.15), we rewrite (2.23) in the form

[TABLE]

Now, formulas (2.18) and (2.24) yield

[TABLE]

Hence, in view of the second equality in (2.3) we have

[TABLE]

which implies (2.21). ∎

Propositions 2.1 and 2.2 lead us to the following theorem.

Theorem 2.3**.**

Let $R$ and $V$ have the form (1.2), let $V_{t}(x,t)$ be a continuous function of $x$ and $t$ combined, and let $R$ and $V$ satisfy MCDE (1.1) in $D$. Assume that three $n\times n$ parameter matrices $A_{1}$, $A_{2}$ and $S(0,0)$ $(\det A_{i}\not=0$, $\,i=1,2;$ $\det S(0,0)\not=0)$, and two $n\times m$ parameter matrices $\Pi_{1}(0,0)$ and $\Pi_{2}(0,0)$ are given, and that the matrix identity (2.1) holds.

Then, $\Pi_{1}(x,t)$, $\Pi_{2}(x,t)$, $S(x,t)$ and $w(x,t,\lambda)$ $($where $w$ is the wave function, i.e., $w(x,t,\lambda)$ satisfies (1.4), (1.5) and $\det w(0,0,\lambda)\not=0)$ are well defined in $D$. Moreover, in the points of invertibility of $S(x,t)$ in $D$, the matrix functions $\widetilde{R}$ and $\widetilde{V}$ given by (2.18) and (2.20), respectively, have the form (1.2) and satisfy (1.1), that is

[TABLE]

The wave function $\widetilde{w}$ $(\det\widetilde{w}(0,0,\lambda)\not=0)$, which corresponds to the transformed MCDE (2.26), is given by the product $w_{A}\,w$:

[TABLE]

Proof.

It follows from [27] that $\Pi_{1}$, $\Pi_{2}$, $S$ and $w$ are well defined. Then, according to (1.4), (1.5) and Proposition 2.1, $\widetilde{w}$ determined by the first equality in (2.28) satisfies the second and third equalities in (2.28), where $\widetilde{G}$ and $\widetilde{F}$ are given by (2.11). Moreover, the second and third equalities in (2.28) imply that the compatibility condition

[TABLE]

holds. Relations (2.11), (2.12) and (2.15) imply that (2.29) holds. Relations (2.11) and (2.19) yield (2.30). Moreover, according to the first equalities in (2.18) and (2.20), the matrix functions $\widetilde{R}$ and $\widetilde{V}$ have the form (2.27).

Taking into account (2.27), (2.29) and (2.30), one can see that $\widetilde{G}$ and $\widetilde{F}$ have the same structure as $G$ and $F$, respectively. Thus, the compatibility condition $\widetilde{G}_{t}-\widetilde{F}_{x}+[\widetilde{G},\widetilde{F}]=0$ yields (2.26) in the same way as (1.3) yields (1.1). ∎

In particular, it is shown in Theorem 2.3 that the Darboux matrix, which transforms the wave function of the initial system into the wave function of the transformed system is given by the transfer function $w_{A}(x,t,\lambda)$.

It is convenient to partition both $\Pi_{1}$ and $\Pi_{2}$ into $n\times m_{1}$ and $n\times m_{2}$ blocks:

[TABLE]

The simplest cases where explicit solutions appear are the cases $V=0$ and $R=I_{m}$, $R=j$, $R=\mathrm{i}I_{m}$ or $R=\mathrm{i}j$. For instance, when $V=0$ and $R=I_{m}$ we obtain $($in view of (2.3)–(2.5)$)$ that

[TABLE]

Example 2.4**.**

Let us consider the case of trivial $V$ and constant diagonal matrix $R$ $(R={\mathcal{D}})$ with the entries $d_{i}$ $($or, written in the block form, blocks ${\mathcal{D}}_{1}$ and ${\mathcal{D}}_{2})$ on the main diagonal$:$

[TABLE]

We set also

[TABLE]

Then relations (2.3)–(2.5), (2.36) and (2.38) yield

[TABLE]

where $\Phi_{i}$ and $\Psi_{i}$ are vector $($row$)$ functions. The function $S(x,t)$ may be recovered from (2.7) and (2.39)–(2.42)$:$

[TABLE]

Using (2.18) and (2.27) we derive

[TABLE]

where $\Phi_{i}$, $\Psi_{i}$ and $S$ are given explicitly in (2.39)–(2.45). Finally, from (2.14), (2.20) and (2.27) we obtain

[TABLE]

and we have a similar formula for $\widetilde{\rho}_{2}$ as well. Clearly, taking into account (2.8), (2.31) and (2.39)–(2.45) we have also an explicit formula for the Darboux matrix $w_{A}$.

2.3 Local matrix dispersionless equations and

asymptotics of the Darboux matrix

Let us set in (1.2)

[TABLE]

Then, MCDE (1.1) takes the form of the local matrix dispersionless equation

[TABLE]

Put in (2.4)–(2.7),

[TABLE]

We will show that GBDT of the initial solutions of (2.49) into the transformed solutions of (2.49) is determined by the triple of matrices $\{A,S(0,0),\Pi(0,0)\}$, where $\det A\not=0$,

[TABLE]

and (2.51) holds. According to (2.3) and (2.47), we have

[TABLE]

It follows from (2.4), (2.5) and from (2.50), (2.53) that

[TABLE]

Equations (2.4)–(2.7) take the form

[TABLE]

Relations (2.51) and (2.56) yield

[TABLE]

Next we show that

[TABLE]

in the case considered in this subsection. In other words, if (2.47) holds for the MCDE solutions, then (2.47) holds for the GBDT-transformed solutions as well. Indeed, relations (2.13), (2.14), (2.50), (2.53), (2.54), and (2.58) imply that

[TABLE]

From the first equalities in (1.2) and (2.20), and from (2.60) we derive the first equality in (2.59). Taking into account (2.54), we rewrite (2.18) in the form

[TABLE]

and the second equality in (2.59) follows. Now, Theorem 2.3 yields the following corollary.

Corollary 2.5**.**

Let the $m_{1}\times m_{2}$, $m_{1}\times m_{1}$ and $m_{2}\times m_{2}$ matrix functions $v(x,t)$, $\rho_{1}(x,t)$ and $\rho_{2}(x,t)$, respectively, satisfy the local matrix dispersionless equation (2.49) and the equalities (2.48), and let $v_{t}(x,t)$ be continuous in $D$. Assume that the parameter matrices $A$ $(\det A\not=0)$, $S(0,0)=S(0,0)^{*}$ and $\Pi(0,0)$ satisfy (2.52).

Then, the matrix functions $\widetilde{v}$, $\widetilde{\rho}_{1}$ and $\widetilde{\rho}_{2}$ given by (2.61) and equalities

[TABLE]

where $S(x,t)$ and $\Pi(x,t)$ are determined by (2.55) and (2.56), satisfy the local matrix dispersionless equation and the equalities $\widetilde{\rho}_{i}(x,t)=\widetilde{\rho}_{i}(x,t)^{*}\quad(i=1,2)$.

The corresponding Darboux matrix $w_{A}$ takes the form

[TABLE]

Further in the subsection, we consider the case

[TABLE]

and study the asymptotics of $v(x,t)$ and $w_{A}(x,t)$ when $x\to\infty$. The asymptotics of $v(x,t)$ and $w_{A}(x,t)$ when $x\to-\infty$ can be studied in the same way.

Formula (2.28) for the fundamental solution $\widetilde{w}$ of the auxiliary systems (for the wave function) takes in this case the form

[TABLE]

where $w_{A}$ is given by (2.64). Hence, the asymptotics of the wave function with respect to $x$ is described by the asymptotics of the Darboux matrix $w_{A}$. Moreover, when we partition $\Pi(0,t)$ into the $n\times m_{1}$ and $n\times m_{2}$ blocks $\Phi_{1}$ and $\Phi_{2}$, we have

[TABLE]

In view of (2.56), under the assumptions

[TABLE]

we have

[TABLE]

When $p=1$ and (2.68) holds, relations (2.56) and (2.57) yield

[TABLE]

Since $S(0,t)>0$, inequalities (2.70) imply that $\det S(x,t)\not=0$ and $\widetilde{v}(x,t)$ is well defined for all $x\in{\mathbb{R}}.$

We note that $w_{A}(4x,t,\lambda)\mathrm{e}^{\mathrm{i}\lambda xj}$ (for each fixed value $t$) is the fundamental solution of the “normalized” Dirac system

[TABLE]

Systems (2.71) as the systems generated by the triples $\{A,S(0,t),\Pi(0,t)\}$ have been studied in a series of papers (see [8, 28, 30] and the references therein). In particular, Weyl functions of the systems (2.71) are rational, and inverse problems to recover systems from the rational Weyl functions have unique and explicit solutions.

Consider the case $p=0$. According to [29, (3.13)] we have

[TABLE]

Without changing $\widetilde{v}$ and $w_{A}$ we may choose $n$, $A$, $S(0,t)>0$, and $\Pi(0,t)$ (see [29]) such that

[TABLE]

where ${\rm Im\ }$ stands for image and the second equality in (2.72) means that the pair $\{A,\Phi_{1}(0,t)\}$ is controllable. Then, we have the following asymptotic relation [29, (3.28)]

[TABLE]

where \varkappa(t)=\lim_{x\to\infty}\Big{(}\mathrm{e}^{-\mathrm{i}xA}S(x,t)\mathrm{e}^{\mathrm{i}xA^{*}}\Big{)}^{-1}, and this limit always exists.

Corollary 2.6**.**

Let $p=0$ and assume that (2.68) holds. Then $\widetilde{v}(x,t)$ does not have singularities when $x\geq 0$, relations (2.72) are valid and $($under a proper choice of $n$, $A$, $S(0,t)>0$, and $\Pi(0,t))$ equality (2.74) holds.

When (2.68) holds (and so $S(0,t)>0$), one can combine [28, Theorem 2.5] (see also the references therein) and [29, Theorem 3.7] in order to obtain the existence of the Jost solutions

[TABLE]

for $\lambda\in{\mathbb{R}}$ and $x\to\infty$. Moreover, from the above-mentioned theorems follows the expression for the reflection coefficient $R_{L}$ of the form

[TABLE]

Corollary 2.7**.**

Let $p=0$ and assume that (2.68) holds. Then the reflection coefficient $R_{L}$ of system (2.76) on the semi-axis $[0,\infty)$ is given by the formula

[TABLE]

where $\theta=A-\mathrm{i}\Phi_{1}(0,t)\Phi_{1}(0,t)^{*}S(0,t)^{-1}$.

Consider the case $p=1$. We already showed that $\widetilde{v}(x,t)$ does not have singularities if (2.68) holds. Moreover, we have [8, Corollary 3.6]

[TABLE]

The asymptotics of $w_{A}$ may be studied in a way similar to the case $p=0$ (see [29]) although the result is somewhat more complicated.

The complex coupled dispersionless equation, which we consider in Section 4, is a scalar subcase of the local matrix dispersionless equation (2.49).

3 GBDT for the nonlocal matrix

dispersionless equations

Recall that the nonlocal matrix dispersionless equations are characterized by the equalities (1.8). Equivalently, the nonlocal matrix dispersionless equations (NMDE) are equations (1.1), where $R$ and $V$ have the form

[TABLE]

In the nonlocal case we assume (3.1) and (3.2) instead of (1.2) and (similar to the subsection 2.3) determine GBDT by 3 parameter matrices. However, these matrices satisfy somewhat different relations. Namely, we set

[TABLE]

so that the identity (2.1) takes the form

[TABLE]

It easily follows from (2.4)–(2.6) that (3.3) and (3.4) yield

[TABLE]

Thus, the identity (2.7) takes the form

[TABLE]

In view of (3.6), we have $X_{0}=\mathrm{i}j^{p}\Pi(-x,t)^{*}S(x,t)^{-1}\Pi(x,t)$ and formula (2.18) takes the form

[TABLE]

Let us again partition $\Pi$ into two blocks: $\Pi=\begin{bmatrix}\Phi_{1}&\Phi_{2}\end{bmatrix}$, where $\Phi_{1}$ is an $m\times m_{1}$ matrix function. Now, (3.2) and (3.8) imply that

[TABLE]

Next, we show that $\widetilde{R}$ given by (2.20) satisfies (under the assumptions of this section) the nonlocal requirement

[TABLE]

Indeed, in view of the relations (2.14), (3.3), and (3.6), we have

[TABLE]

From the last equality in (2.3) and the formulas (3.1) and (3.2) we derive

[TABLE]

Formulas (2.13), (3.16) and (3.17) imply that

[TABLE]

Finally, the first equality in (2.20) and formula (3.18) yield (3.13).

Rewriting (2.4), (2.6), and (2.8) under assumptions of this section, we obtain

[TABLE]

Recall that in this section we assume that the relations (3.3) and (3.4) hold (in particular, GBDT is determined by the triple $\{A,S(0,0),\Pi(0,0)\}$). Now, we can rewrite Theorem 2.3 for the NMDE case.

Theorem 3.1**.**

Let $R$ and $V$ have the forms (3.1) and (3.2), respectively, let $V_{t}(x,t)$ be continuous in $D$, and let $R$ and $V$ satisfy (1.1) in $D$. Assume that two $n\times n$ parameter matrices $A$ and $S(0,0)$ and one $n\times m$ parameter matrix $\Pi(0,0)$ are given, and that (3.5) holds.

Then, $\Pi(x,t)$, $S(x,t)$ and $w(x,t,\lambda)$ $($where $w$ is the wave function, i.e., $w(x,t,\lambda)$ satisfies (1.4), (1.5) and $\det w(0,0,\lambda)\not=0)$ are well defined in $D$.

Moreover, in the points of invertibility of $S(x,t)$ in $D$, the matrix function $\widetilde{R}$ given by (2.20) $($where $X_{-1}$ and $Y_{-1}$ have the forms (3.14) and (3.15)$)$ and the matrix function $\widetilde{V}$ given by (3.8) satisfy NCDE, that is, the relations (3.13) and (3.11) are valid and the equations

[TABLE]

are satisfied.

The wave function $\widetilde{w}$ $(\det\widetilde{w}(0,0,\lambda)\not=0)$, which corresponds to the transformed NCDE (3.22), is given by the product $w_{A}\,w$, where the Darboux matrix $w_{A}$ has the form (3.21). In other words, the relations (2.28)–(2.30) are valid.

4 GBDT for the complex

coupled dispersionless equations

Recall that in order to obtain the complex coupled dispersionless equations (CCDE)

[TABLE]

we set in the MCDE (see (1.1) and (1.2)) the equalities (1.7). In particular, since $m_{1}=m_{2}=1$, the functions $\rho$ and $v$ are scalar functions and we rewrite (2.3) and (1.2) in the form

[TABLE]

In view of (4.3), the coefficients $q_{1},\,q_{0},\,Q_{1}$ given by (4.2) have the property

[TABLE]

In order to construct GBDT for the CCDE equations (4.1), we set in the GBDT for MCDE in Section 2 the equalities

[TABLE]

and $S(0,0)=S(0,0)^{*}$. It means that GBDT is determined by 3 parameter matrices:

[TABLE]

In view of (4.5), we rewrite (2.4) in the form

[TABLE]

Taking into account (2.5) and (4.4)–(4.6), we see that

[TABLE]

Hence, the identity (2.7) takes the form

[TABLE]

The matrix function $S(x,t)=S(x,t)^{*}$ is determined now by $S(0,0)$ and the equations

[TABLE]

which follow from (2.6).

The GBDT-transformed solution $\{\widetilde{R},\,\widetilde{V}\}$, Darboux matrix $w_{A}$ and wave function $\widetilde{w}$ are expressed via $A$, $\Pi(x,t)$ and $S(x,t)$. Let us show that $\widetilde{R}$ and $\widetilde{V}$ have the form (4.3):

[TABLE]

and so $\widetilde{\rho}$ and $\widetilde{v}$ satisfy CCDE. Indeed, $X_{0}$, $X_{-1}$ and $Y_{-1}$ (given by (2.14)) take now the form

[TABLE]

Thus, we rewrite (2.18) as

[TABLE]

According to (4.12), $\widetilde{V}$ has the form (4.9), where

[TABLE]

According to (2.13), (2.15) and (4.2) we have

[TABLE]

where ${\mathrm{tr}}$ stands for trace. In view of (2.20) and (4.14), the first equality in (4.9) holds.

It remains to prove that $\widetilde{\rho}=\overline{\widetilde{\rho}}$. From (4.11) we see that

[TABLE]

Hence, the last equality in (4.4) and the equalities in (2.20) imply that $\widetilde{Q}_{-1}^{*}=-j^{p+1}\widetilde{Q}_{-1}j^{p+1}$ and so $\widetilde{R}=\widetilde{R}^{*}$. That is, we have

[TABLE]

which finishes the proof of (4.9). We obtained the following corollary of Theorem 2.3.

Corollary 4.1**.**

Let $v_{t}(x,t)$ be continuous in $D$, and let the functions $\rho$ and $v$ satisfy CCDE (4.1) in $D$. Assume that two $n\times n$ parameter matrices $A$ and $S(0,0)=S(0,0)^{*}$ and one $n\times 2$ parameter matrix $\Pi(0)$ are given, and that the relations

[TABLE]

hold. Introduce $\Pi(x,t)$ and $S(x,t)$ using (4.6), (4.8) \big{(}and (4.2), (4.3), where p=(1+\varkappa)/2\big{)}.

Then, in the points of invertibility of $S(x,t)$ in $D$, the functions $\widetilde{\rho}$ $($given by (4.15) and (4.11)$)$ and $\widetilde{v}$ $($given by (4.13)$)$ satisfy CCDE $:$

[TABLE]

Moreover, a wave function $w(x,t,\lambda)$ $($where $\det w(0,0,\lambda)\not=0)$ is well defined in $D$ via (4.3) and auxiliary systems

[TABLE]

The wave function $\widetilde{w}$ $(\det\,\widetilde{w}(0,0,\lambda)\not=0)$, which corresponds to the transformed CCDE (4.17), is given by the product $\widetilde{w}=w_{A}\,w$, where the Darboux matrix $w_{A}$ has the form

[TABLE]

Example 4.2**.**

In order to present an example of the solution of CCDE (4.1), we set in Example 2.4 $($in accordance with (1.7) and (4.5)$)$

[TABLE]

$\Psi_{1}(0,0)=-\mathrm{i}\Phi_{1}(0,0),$* and $\Psi_{2}(0,0)=(-1)^{p}\mathrm{i}\Phi_{1}(0,0)$. For simplicity of notations, we put $\Phi_{i}(0,0)=c_{i}$. In view of (2.43)–(2.45) we have*

[TABLE]

and $S(x,t)=\overline{S(x,t)}$. The formula for $\widetilde{v}_{1}$ in Example 2.4 takes the form

[TABLE]

Finally, formula (2.46) takes the form

[TABLE]

5 Coupled dispersionless equations

Similarly to Section 4, we consider here the case of scalar function $\rho$ (and scalar $v_{1}$ and $v_{2}$). Setting in (1.1)

[TABLE]

we rewrite (1.1) in the form

[TABLE]

which is equivalent, for instance, to [20, (1.2)] (see also the references therein). The following corollary of Theorem 2.3 is valid.

Corollary 5.1**.**

Let the conditions of Theorem 2.3 hold and assume additionally that

[TABLE]

Then, $\widetilde{v}_{1}$ and $\widetilde{v}_{2}$ given by (2.18), and $\widetilde{\rho}$ given by the formula

[TABLE]

satisfy equations (5.2), that is,

[TABLE]

Proof.

Taking into account (1.1), (1.2) and (2.20), the only fact which we need to prove is that $\widetilde{R}$ has the form $\widetilde{\rho}I_{2}$, that is, that $\widetilde{\rho}_{1}=\widetilde{\rho}_{2}$. Similar to the calculation in the previous section, relations (2.3) and (2.15) yield

[TABLE]

and the equality $\widetilde{\rho}_{1}=\widetilde{\rho}_{2}$ follows from (2.20) and (5.6). ∎

For the nonlocal situation

[TABLE]

equations (5.5) have the form

[TABLE]

In other words, under conditions (5.3) and (5.7) system (1.1) is equivalent to the system (5.8), (5.9). (Note that $\rho$ and $v$ are scalar functions.)

Assume further that the relations (3.3)–(3.5) hold (in particular, GBDT is determined by the triple $\{A,S(0,0),\Pi(0,0)\}$). Below, we formulate a corollary of Theorem 3.1.

Corollary 5.2**.**

Let $v_{t}$ be continuous in $D$, assume that $\overline{\rho(x,t)}=-\rho(-x,t)$, and let $v$ and $\rho$ satisfy (5.8), (5.9) in $D$.

Then, $\widetilde{v}$ given by (3.12) and $\widetilde{\rho}$ given by (5.4) satisfy (5.8), (5.9) $($as well as $v$ and $\rho)$, that is, the following equalities hold$:$

[TABLE]

We also have $\overline{\widetilde{\rho}(x,t)}=-\widetilde{\rho}(-x,t)$.

Proof.

Similar to the Corollary 5.1 we need only to prove that $\widetilde{\rho}_{1}=\widetilde{\rho}_{2}$ (after which we may use Theorem 3.1). The equality $\widetilde{\rho}_{1}=\widetilde{\rho}_{2}$ follows from (2.20) and (5.6). ∎

6 Examples and figures

In these examples we construct explicit solutions of the nonlocal equations (5.10), (5.11). We set

[TABLE]

We put

[TABLE]

which corresponds [29] to the simplest case of the Weyl function $($reflection coefficient$)$ with a pole of the order more than one $($so called multipole case$)$. For the literature on the multipole cases see, for instance, [24, 33] and the references therein.

Using notations $\Phi_{i}(0,0)=C_{i}=\begin{bmatrix}c_{i1}\\ c_{i2}\end{bmatrix}$, we $($similar to the deduction of (2.32)$)$ obtain

[TABLE]

where $A_{0}=\begin{bmatrix}0&1\\ 0&0\end{bmatrix}$. In the same way as (6.14), we derive

[TABLE]

Relations (6.14)–(6.15) provide an explicit expression for

[TABLE]

Next, using (3.7) we easily express $S(x,t)=\{S_{i\ell}(x,t)\}_{i,\ell=1}^{2}$ in terms of $K(x,t):$

[TABLE]

Finally, from (3.12), (5.4) and (6.12) it follows that

[TABLE]

Recall that $($according to Corollary 5.2$)$ $\widetilde{v}$ and $\widetilde{\rho}$ satisfy (5.10), (5.11).

The fundamental solution $($wave function$)$ $w(x,t,\lambda)$ of the initial systems (1.4) and (1.5), where $V=0$ and $R=\mathrm{i}I_{2}$ is given by the formula

[TABLE]

In view of (6.13)–(6.20) we have explicit formulas for the Darboux matrix $w_{A}(x,t,\lambda)=I_{2}-\mathrm{i}j^{p}\Pi(-x,t)^{*}S(x,t)^{-1}(A-\lambda I_{2})^{-1}\Pi(x,t)$. Thus the wave function $w_{A}(x,t,\lambda)w(x,t,\lambda)$ of the transformed system with $\widetilde{v}$ and $\widetilde{\rho}$ given by (6.21) and (6.22), respectively, is also expressed explicitly.

Let us consider several explicit formulas in greater detail. In the following we set $a_{1}=\Re(a)$ and $a_{2}=\Im(a)$.

Case 1. The simplest case is the case where $c_{12}=c_{21}=0$, that is, $C_{1}=\begin{bmatrix}c_{11}\\ 0\end{bmatrix}$ and $C_{2}=\begin{bmatrix}0\\ c_{22}\end{bmatrix}$. Here, relations (6.14)–(6.22) after some calculations yield:

[TABLE]

In particular, for

[TABLE]

the behaviour of $|\widetilde{v}|$ and $\ln|\widetilde{\rho}|$ is shown on Figure 1.

Case 2. When $p=1$, $C_{1}=\begin{bmatrix}c_{11}\\ c_{12}\end{bmatrix}$ and $C_{2}=\begin{bmatrix}0\\ c_{22}\end{bmatrix}$ $(c_{11},\,c_{12},\,c_{22}\in{\mathbb{R}})$, our choice of non-diagonal $A$ leads to polynomials

[TABLE]

(in addition to the exponents) in the formulas for $\widetilde{v}$ and $\widetilde{\rho}$. Namely, we have:

[TABLE]

The behaviour of $\widetilde{v}$ and $\widetilde{\rho}$ is in this case more complicated, see Figure 2, where

[TABLE]

In some other cases the formulas are more complicated and we restrict ourselves to figures only. See Figure 3, where

[TABLE]

see Figure 4, where

[TABLE]

and see Figure 5, where

[TABLE]

Acknowledgments. This research was supported by the Austrian Science Fund (FWF) under Grant No. P29177.