On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

Phan Van Tin

doi:10.3934/eect.2021028

Article Contents

2022, Volume 11, Issue 3: 837-867. Doi: 10.3934/eect.2021028

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On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

Phan Van Tin

Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

Received: January 2021

Revised: April 2021

Published: June 2022

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Abstract

In this paper we consider the Schrödinger 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( \mathbb{R}) $ and in $ \phi+H^k( \mathbb{R}) $. Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.

Keywords:

Nonlinear derivative Schrödinger equations,
Cauchy problem,
nonvanishing boundary condition.

Mathematics Subject Classification: 35Q55, 35A01.

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