Construction of the multi-soliton trains for a generalized derivative nonlinear Schr\''odinger equations by a fixed point method

TL;DR

This paper proves the existence of multi-soliton train solutions for a generalized derivative nonlinear Schrödinger equation with nonlinearity parameter greater than 2, using fixed point methods and Strichartz estimates.

Contribution

It extends the construction of multi-soliton trains to the case where the nonlinearity parameter exceeds 2, which was not previously established.

Findings

Existence of multi-soliton trains for σ > 2 in energy space.

Application of fixed point arguments around the soliton profiles.

Use of Strichartz estimates to facilitate the proofs.

Abstract

We consider a derivative nonlinear Schr{\"o}dinger equation with general nonlinearlity: i$\partial$tu + $\partial$ 2 x u + i|u| 2$\sigma$ $\partial$xu = 0, In [12], the authors prove the stability of two solitary waves in energy space for $\sigma$ $\in$ (1, 2). As a consequence, there exists a two-soliton trains in energy space for $\sigma$ $\in$ (1, 2). Our goal in this paper is proving the existence of multi-soliton trains in energy space for $\sigma$ 5 2. Our proofs proceed by xed point arguments around the desired prole, using Strichartz estimates.