Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations

TL;DR

This paper extends the construction of multi-soliton solutions to the L2 supercritical gKdV and NLS equations, employing a topological approach to manage instability directions, thus broadening the understanding of soliton dynamics.

Contribution

It introduces a novel topological method to construct multi-soliton solutions in the L2 supercritical regime for gKdV and NLS equations, which was not previously achieved.

Findings

Successfully constructed multi-soliton solutions in the supercritical case

Demonstrated control of instability directions using topological arguments

Extended known results from critical/subcritical to supercritical regimes

Abstract

Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as $t\to +\infty$, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.