Open AccessOn the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions
Author(s) -
Phan van Tin
Publication year - 2022
Publication title -
evolution equations and control theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 19
eISSN - 2163-2480
pISSN - 2163-2472
DOI - 10.3934/eect.2021028
Subject(s) - mathematics , initial value problem , cauchy problem , nonlinear system , cauchy distribution , nonlinear schrödinger equation , mathematical analysis , schrödinger equation , mathematical physics , pure mathematics , physics , quantum mechanics
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 \begin{document}$ X^k( \mathbb{R}) $\end{document} and in \begin{document}$ \phi+H^k( \mathbb{R}) $\end{document} . Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.