On the Cauchy problem for a nonlocal nonlinear Schrödinger equation

Hongwei Wang; Amin Esfahani

doi:10.3934/dcdsb.2022039

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2022, Volume 27, Issue 12: 7185-7206. Doi: 10.3934/dcdsb.2022039

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On the Cauchy problem for a nonlocal nonlinear Schrödinger equation

Hongwei Wang^1, and
Amin Esfahani^2, ,

1.
School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China
2.
Department of Mathematics, Nazarbayev University, Nur-Sultan 010000, Kazakhstan

^*Corresponding author: Amin Esfahani
^*Corresponding author: Amin Esfahani

Received: July 31, 2021

Revised: December 31, 2021

Early access: March 2022

Published: December 2022

H. W. was supported by Key Project of Natural Science Foundation of Educational Committee of Henan Province(No. 20A110007). A. E. was supported by the Social Policy Grant (SPG) funded by Nazarbayev University, Kazakhstan

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Abstract

This paper considers the one-dimensional Schrödinger equation with nonlocal nonlinearity that describes the interactions of nonlinear dispersive waves. We obtain some the local well-posedness and ill-posedness result associated with this equation in the Sobolev spaces. Moreover, we prove the existence of standing waves of this equation. As corollary, we derive the conditions under which the solutions are uniformly bounded in the energy space.

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Mathematics Subject Classification: Primary: 35A15, 35Q55; Secondary: 35A01.

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Figure 1. The graphs of $ K $ with negative values of $ \beta $ are shown in the left and in the right figures with both signs

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Figure 2. Plots of ground states of (1) with $ \omega = 1 $ and various values of $ \beta $

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