Multi-Speeds solitary waves solutions for nonlinear Schr\"odinger systems

TL;DR

This paper proves the existence of multi-speeds solitary wave solutions for nonlinear Schrödinger systems, showing they behave as multiple scalar solitary waves traveling at different speeds over time.

Contribution

It introduces and constructs a new class of solutions called multi-speeds solitary waves for nonlinear Schrödinger systems, using backward-in-time approximation and energy estimates.

Findings

Existence of multi-speeds solitary wave solutions.

Solutions behave as multiple solitary waves at different speeds.

Construction method based on backward time approximation.

Abstract

We prove the existence of a new type of solutions to a nonlinear Schr\"odinger system. These solutions, which we call "multi-speeds solitary waves", are behaving at large time as a couple of scalar solitary waves traveling at different speeds. The proof relies on the construction of approximations of the multi-speeds solitary waves by solving the system backwards in time and using energy methods to obtain uniform estimates.