Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. reflection

TL;DR

This paper analyzes the behavior of soliton-like solutions in slowly varying gKdV equations, showing they can be refracted or reflected depending on initial energy, and introduces a new type of solution that is not pure at infinity.

Contribution

It provides a nearly complete description of soliton behaviors in slowly varying gKdV equations, including a novel reflection phenomenon and non-pure solutions at infinity.

Findings

Solitons can be refracted or reflected based on initial energy.

Introduction of a new soliton-like solution not pure at infinity.

Extension of known behaviors to a broader class of gKdV equations.

Abstract

In this work we continue the description of soliton-like solutions of some slowly varying, subcritical gKdV equations. In this opportunity we describe, almost completely, the allowed behaviors: either the soliton is refracted, or it is reflected by the potential, depending on its initial energy. This last result describes a new type of soliton-like solution for gKdV equations, also present in the NLS case. Moreover, we prove that the solution is not pure at infinity, unlike the standard gKdV soliton.