Soliton solutions and dynamical evolutions of a generalized AKNS system in the framework of inverse scattering transform

Abstract

It is important and interesting to derive integrable systems or construct soliton solutions of nonlinear partial differential equations (PDEs). In this paper, a new and more general Ablowitz–Kaup–Newell–Segur (AKNS) system with Lax integrability is derived and solved in the framework of inverse scattering transform (IST). To be specific, the known AKNS’s linear spectral problem and its time evolution equation are first generalized by embedding a nonisospectral parameter whose varying with time obeys the rational function of spectral parameter. Based on the generalized AKNS spectral problem and time evolution equation, we then derive a generalized AKNS system with infinite number of terms. Finally, in the case of reflectionless potentials, explicit n-soliton solutions of the generalized AKNS system are formulated through the IST method. It is shown that the dynamical evolutions of the obtained one-soliton solutions and two-soliton solutions possess time-varying speeds and amplitudes in the process of propagations.