Integrability and asymptotic behaviour of a differential-difference matrix equation
Introduction
The aim of the present paper is to undertake a study of the nonautonomous lattice as well as of its autonomous limit 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 () 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 is known to be a completely integrable equation. The transformation maps solutions of (1.3) to solutions of the equation Alternatively, we may regard the system 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 using the transformation 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.
Section snippets
Integrability of the autonomous matrix lattice
We start by considering the autonomous matrix equation (1.2), that is It can be shown that solutions of this equation are mapped by the transformation to solutions of the equation As with the scalar case (1.6), we may regard the system as a BT between Eqs. (2.1), (2.3). Let the shift () and difference () operators be defined by so for any integer and . The matrix
The nonautonomous matrix lattice
In this section we give results for the nonautonomous matrix lattice (1.1). These results are analogous to those given in the previous section for the autonomous case. In addition, we show how the autonomous case may be derived from the nonautonomous case via a limiting process, and also briefly discuss results for the scalar case of the nonautonomous lattice.
Conclusions
We have considered the matrix lattice equation , in both its autonomous () and nonautonomous () forms. For each of these cases we have proved integrability by constructing a Miura map to a corresponding integrable equation in , that is, Eqs. (2.3), (3.3) respectively. We have also sought transformations from these equations in to corresponding autonomous and nonautonomous matrix Volterra systems, in two matrix dependent variables and . However, in
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors are grateful to the Ministry of Economy and Competitiveness of Spain for supporting their work through contract MTM2016-80276-P (AEI/FEDER, EU).
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