On The Cauchy Problem For A Derivative Nonlinear Schr{\"o}Dinger Equation With Nonvanishing Boundary Conditions

TL;DR

This paper studies a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions, establishing local well-posedness, conservation laws, and explicit stationary solutions within specific functional spaces.

Contribution

It introduces the analysis of this Schrödinger equation with nonvanishing boundaries, providing new results on well-posedness, conservation laws, and stationary solutions.

Findings

Local well-posedness on Zhidkov spaces and in $\

ext{existence of conservation laws using localizing functions}

ext{explicit formulas for stationary solutions}

Abstract

In this paper we consider the Schr{\"o}dinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on Zhidkov spaces X k (R) and in $\phi$ + H k (R). Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.