Cauchy matrix structure of the Mel’nikov model of long–short wave interaction

Abstract

We propose a systematic method to construct the Mel’nikov model of long–short wave interactions, which is a special case of the Kadomtsev–Petviashvili (KP) equation with self-consistent sources (KPSCS). We show details how the Cauchy matrix approach applies to Mel’nikov's model which is derived as a complex reduction of the KPSCS. As a new result we find that in the dispersion relation of a 1-soliton there is an arbitrary time-dependent function that has previously not reported in the literature about the Mel’nikov model. This function brings time variant velocity for the long wave and also governs the short-wave packet. The variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated.

Nonisospectral effects on generating localized waves

“…the nKdV equation (4), the nonisospectral complex mKdV equation (20) and the nonisospectral Hirota equation ( 23) as examples, and derived their solutions in terms of Wronskians and double Wronskians. Since these equations are integrable, other approaches, such as the inverse scattering transform [34] and Cauchy matrix approach [35][36][37][38] can also be employed to get their solutions. Note that the later two equations belong to the ZS-AKNS hierarchy and we derived their solutions from those of unreduced systems (20) and ( 23) by means of recently developed reduction technique [32,33], which has been demonstrated effective in practice in getting solutions of the reduced equations involved complex reductions (e.g.…”

The Integrability, Equivalence and Solutions of Two Kinds of Integrable Deformed Fourth-Order Matrix Nls Equations

“…function in the singular dispersion relation. Further, we proved that FMNLSSCS (3) and FMNLS-MB (4) are equivalent and proposed the explicit expression of the N-soliton solution via the Cauchy matrix method [25,26].…”

Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equations

“…It is first systematically introduced in [20] to investigate integrable quadrilateral equations and later developed in [21,22] to more general cases. It has also been applied to the Zakharov-Shabat-Ablowitz-Kaup-Newell-Segur (ZS-AKNS) system [23], equations with self-consistent sources [24], and the self-dual Yang-Mills equation [25,26], etc. The purpose of this paper is not only to construct solutions to the three NNLSEs in (1.2), but also to extend the Cauchy matrix approach to the non-isospectral case, as the Sylvester-type equation in the Cauchy matrix scheme of the ZS-AKNS system is a typical type (see [22,27] for the Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) type equations).…”