Integrability and asymptotic behaviour of a differential-difference matrix equation

Highlights

• We give Miura maps for autonomous and nonautonomous matrix lattice equations and systems.

• We prove the integrability of an autonomous and also of a nonautonomous matrix lattice.

• We obtain integrable multicomponent autonomous and nonautonomous lattices and Miura maps.

• We give a new integrable nonautonomous matrix Volterra equation along with its Lax pair.

• We give asymptotic reductions of matrix lattices to matrix potential KdV and matrix KdV.

Abstract

In this paper we consider the matrix lattice equation ${\mathbf{U}}_{n,t}({\mathbf{U}}_{n+1}-{\mathbf{U}}_{n-1})=g\left(n\right)\mathbf{I}$, in both its autonomous ($g\left(n\right)=2$) and nonautonomous ($g\left(n\right)=2n-1$) forms. We show that each of these two matrix lattice equations are integrable. In addition, we explore the construction of Miura maps which relate these two lattice equations, via intermediate equations, to matrix analogs of autonomous and nonautonomous Volterra equations, but in two matrix dependent variables. For these last systems, we consider cases where the dependent variables belong to certain special classes of matrices, and obtain integrable coupled systems of autonomous and nonautonomous lattice equations and corresponding Miura maps. Moreover, in the nonautonomous case we present a new integrable nonautonomous matrix Volterra equation, along with its Lax pair. Asymptotic reductions to the matrix potential Korteweg–de Vries and matrix Korteweg–de Vries equations are also given.

Introduction

The aim of the present paper is to undertake a study of the nonautonomous lattice

$${\mathbf{U}}_{n,t}=(2n-1){({\mathbf{U}}_{n+1}-{\mathbf{U}}_{n-1})}^{-1},$$

as well as of its autonomous limit

$${\mathbf{U}}_{n,t}=2{({\mathbf{U}}_{n+1}-{\mathbf{U}}_{n-1})}^{-1}.$$

Eq. (1.1) may be derived using the auto-Bäcklund transformations (aBTs) of a certain matrix partial differential equation (PDE), or alternatively those of a matrix second Painlevé ($P_{II}$) equation [1] (see also [2]). The limiting process from (1.1) to (1.2) is presented later.

The scalar case of the autonomous equation (1.2), that is

$$u_{n,t}=\frac{2}{u_{n+1}-u_{n-1}},$$

is known to be a completely integrable equation. The transformation

$$w_n=\frac{1}{u_{n+1}-u_{n-1}},$$

maps solutions of (1.3) to solutions of the equation

$$w_{n,t}=-2w_n^2(w_{n+1}-w_{n-1}).$$

Alternatively, we may regard the system

$$w_n=\frac{1}{u_{n+1}-u_{n-1}},u_{n,t}=2w_n,$$

as a Bäcklund transformation (BT) between (1.3), (1.5). Solutions of Eq. (1.5) may in turn be mapped to solutions of the (rescaled) Volterra lattice

$$y_{n,t}=-2y_n(y_{n+1}-y_{n-1}),$$

using the transformation

$$y_n=w_n{w}_{n+1}.$$

Eq. (1.5) and the Volterra equation (1.7) are well-known completely integrable equations. The above three scalar Eqs. (1.3), (1.5), (1.7), results on their integrability, relations between them as well as to other known equations, and also various generalisations thereof, can be found in [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. In particular, the Miura transformation from (1.5) to (1.7), and the composite Miura transformation from (1.3) to (1.7), can be found for example in [13], [18]. We recall that in [18] the integrability of scalar lattices is shown by constructing Miura transformations to known integrable cases. For example, the existence of a Miura transformation from (a generalisation of) (1.3) to the integrable Volterra equation (1.7) is used to prove the integrability of the former.

One aim of the present paper is to consider to what extent it is possible to extend the above chain of transformations for the scalar autonomous lattice (1.3) to the matrix case (1.2). Here our objective is two-fold: to establish the integrability of (1.2) by extending the mapping (1.4) to the matrix case; and to obtain matrix generalisations of Eq. (1.7) and the mapping (1.8). A second aim is to consider asymptotic reductions of autonomous matrix lattices to matrix PDEs. These topics are considered in Section 2. In Section 3 we turn our attention to the nonautonomous matrix lattice (1.1), and study the possible extension of the above chain of transformations to this case — with the same two-fold objective as in the autonomous case — as well as the asymptotic reduction of nonautonomous matrix lattices to matrix PDEs, and the derivation of autonomous lattices as limiting cases of nonautonomous lattices. We also briefly consider results in the nonautononmous scalar case. In each of Sections 2 The autonomous matrix lattice, 3 The nonautonomous matrix lattice a variety of new results are presented. The final section of the paper consists of a discussion and conclusions.