Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equations

Abstract

The paper aims to develop a direct approach, namely, the Cauchy matrix approach, to non-isospectral integrable systems. In the Cauchy matrix approach, the Sylvester equation plays an central role, which defines a dressed Cauchy matrix to provide τ functions for the investigated equations. In this paper, using the Cauchy matrix approach, we derive three non-isospectral nonlinear Schrödinger
equations and their explicit solutions. These equations are generically related to time-dependent spectral parameter in the Zakharov-Shabat-Ablowitz-Kaup-Newell-Segur spectral problem. Their solutions are obtained from the solutions of unreduced non-isospectral nonlinear Schrödinger equations through complex reduction. These solutions are analyzed and illustrated to show the non-isospectral effects in dynamics of solitons.

Cauchy matrix approach for H1 a equation in the torqued Adler–Bobenko–Suris lattice list

“…In this paper, we would like to construct multisoliton solutions and Jordan-block solutions for the H1 a equation (1.3) by utilizing the Cauchy matrix approach [12,13], where the method arose from the well-known Sylvester equation in matrix theory [20] and could be viewed as a byproduct of the direct linearization method [7,21]. The idea behind the Cauchy matrix approach, to use the determining equation set (DES) as the starting point, was developed further in a series of papers [22][23][24][25][26][27][28][29]. The paper is organized as follows.…”