Well-posedness for a modified nonlinear Schrodinger equation modeling the formation of rogue waves

TL;DR

This paper proves the well-posedness of a modified nonlinear Schrödinger equation modeling rogue wave formation, using advanced analytical techniques in Sobolev and Bourgain spaces.

Contribution

It establishes well-posedness for a higher order MNLS with novel microlocal analysis and a new trilinear estimate, extending understanding of rogue wave models.

Findings

Well-posedness in Sobolev spaces with exponent ≥ 0.

Use of Bourgain spaces adapted to the linear symbol.

Application of microlocal analysis and a new trilinear estimate.

Abstract

The Cauchy problem for a higher order modification of the nonlinear Shcrodinger equation (MNLS) on the line is shown to be well-posed in Sobolev spaces with exponent $\ge 0$. This result is achieved by demonstrating that the associated integral operator is a contraction on a Bourgain space that has been adapted to the particular linear symbol present in the equation. the ctraction is proved by using microlocal analysis and a new trilinear estimate.