Variational Approach to studying solitary waves in the nonlinear Schrodinger equation with Complex Potentials

TL;DR

This paper develops a variational method to analyze solitary wave interactions with complex potentials in the nonlinear Schrödinger equation, providing analytical and numerical insights into their dynamics and stability.

Contribution

It introduces a four-parameter variational approximation for the NLSE with complex potentials, offering analytical results and comparisons with existing methods.

Findings

Good agreement with direct numerical simulations

Analytical expressions for small oscillation frequencies

Identification of instability conditions

Abstract

We discuss the behavior of solitary wave solutions of the nonlinear Schr{\"o}dinger equation (NLSE) as they interact with complex potentials, using a four parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope of $dp(t)/dv(t)$ becomes negative when plotted parametrically as a function of time, where $p(t)$ is the momentum of the solitary wave and $v(t)$ the velocity.