Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach

TL;DR

This paper uses Melnikov's perturbation method to analytically identify conditions for the existence of stationary solitary waves in complex potentials, regardless of symmetry, and verifies results with numerical simulations.

Contribution

It introduces a general analytical framework for solitary wave existence in non-symmetric complex potentials using Melnikov theory.

Findings

Analytical conditions for solitary wave existence derived

Constraints on real and imaginary parts of potentials identified

Numerical simulations confirm analytical predictions and explore wave dynamics

Abstract

The existence of stationary solitary waves in symmetric and non-symmetric complex potentials is studied by means of Melnikov's perturbation method. The latter provides analytical conditions for the existence of such waves that bifurcate from the homogeneous nonlinear modes of the system and are located at specific positions with respect to the underlying potential. It is shown that the necessary conditions for the existence of continuous families of stationary solitary waves, as they arise from Melnikov theory, provide general constraints for the real and imaginary part of the potential, that are not restricted to symmetry conditions or specific types of potentials. Direct simulations are used to compare numerical results with the analytical predictions, as well as to investigate the propagation dynamics of the solitary waves.