Well-posedness for one-dimensional derivative nonlinear Schr\"odinger equations

TL;DR

This paper proves local well-posedness for a class of one-dimensional derivative nonlinear Schrödinger equations with high nonlinearity, using gauge transformation and Littlewood-Paley techniques, for initial data in the fractional Sobolev space.

Contribution

It establishes the local well-posedness of the derivative nonlinear Schrödinger equation for any real $k extgreater= 5$ in $H^{1/2}$, extending previous results to higher nonlinearities.

Findings

Proves local well-posedness in $H^{1/2}$ for $k extgreater= 5$

Uses gauge transformation and Littlewood-Paley decomposition

Handles equations with non-zero real parameter $ extlambda$

Abstract

In this paper, we investigate the one-dimensional derivative nonlinear Schr\"odinger equations of the form $iu_t-u_{xx}+i\lambda\abs{u}^k u_x=0$ with non-zero $\lambda\in \Real$ and any real number $k\gs 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.