Soliton robustness, interaction and stability for a variable coefficients Schrödinger(VCNLS) equation with inverse scattering transformation

The inverse scattering transformation(IST) method is an important method for solving soliton equations. In this paper, some novel solutions of a variable coefficients Schrödinger(VCNLS) equation are obtained with IST method. And we find that these waves have not the effected by other external waves, then call them on robustness waves. The VCNLS equation is derived from a non-isospectral AKNS equation hierarchy, and the spectral parameter is determined by an ordinary differential equation with polynomial nonlinearity. The VCNLS equation with the periodic and harmonic external potentials is solved via the inverse scattering transformation(IST) method, which possess the parity-time(PT) symmetric invariance. Unlike the classical NLS equation, the characteristic functions of the CVNLS equation with space–time external potential have two different symmetric reductions and two different backscattering solutions, which extends the application of IST and is helpful for the CVNLS equation. Some novel bright soliton solutions are obtained, which are different to the previous results in the classical case. The stabilities of the bright soliton solutions and the effect of the phase noise on the solutions are analyzed in a wide region through numerically. Especially, the interaction dynamics of between two solitons are investigated through addressing numerically, which have not a effected by other external waves. Thus, the novel features are different from these usual features of solitons, which have not a effected by other external waves.