Multi-soliton solutions for the supercritical gKdV equations

TL;DR

This paper constructs and classifies multi-soliton solutions for the supercritical gKdV equations, extending previous results from subcritical and critical cases to the supercritical regime.

Contribution

It introduces an N-parameter family of multi-solitons for supercritical gKdV and proves that all multi-solitons belong to this family, providing a complete classification.

Findings

Constructed an N-parameter family of multi-solitons for supercritical gKdV.

Proved that all multi-solitons are contained within this family.

Extended the classification of multi-solitons to the supercritical case.

Abstract

For the L^2 subcritical and critical (gKdV) equations, Martel proved the existence and uniqueness of multi-solitons. Recall that for any N given solitons, we call multi-soliton a solution of (gKdV) which behaves as the sum of these N solitons asymptotically as time goes to infinity. More recently, for the L^2 supercritical case, Cote, Martel and Merle proved the existence of at least one multi-soliton. In the present paper, as suggested by a previous work concerning the one soliton case, we first construct an N-parameter family of multi-solitons for the supercritical (gKdV) equation, for N arbitrarily given solitons, and then prove that any multi-soliton belongs to this family. In other words, we obtain a complete classification of multi-solitons for (gKdV).