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.
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