Localized nonlinear waves in systems with time- and space-modulated nonlinearities

TL;DR

This paper develops a method to find explicit solutions for nonlinear Schrödinger equations with time- and space-dependent nonlinearities, revealing complex wave phenomena like breathers and oscillating solitons, with implications for matter-wave physics.

Contribution

It introduces a general theory using similarity transformations to construct explicit solutions for variable-coefficient nonlinear Schrödinger equations.

Findings

Derived explicit solutions including breathers and oscillating solitons.

Demonstrated the applicability to matter-wave systems.

Provided a framework for analyzing nonlinear wave dynamics with variable parameters.

Abstract

Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schr\"odinger equations with potentials and nonlinearities depending on time and on the spatial coordinates. We present the general theory and use it to calculate explicitly non-trivial solutions such as periodic (breathers), resonant or quasiperiodically oscillating solitons. Some implications to the field of matter-waves are also discussed.