Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equations

Abstract

This 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 a 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 the 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.