On nonlinear effects in multiphase WKB analysis for the nonlinear Schrödinger equation *

IF 1.6 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity Pub Date : 2024-05-12 DOI:10.1088/1361-6544/ad4505

Rémi Carles

引用次数: 0

Abstract

We consider the Schrödinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that like in the linear case, a superposition principle holds on some time interval independent of the semiclassical parameter, in several régimes in term of the size of initial data with respect to the semiclassical parameter. When nonlinear effects are strong in terms of the semiclassical parameter, we invoke properties of compressible Euler equations. For weaker nonlinear effects, we show that there may be no nonlinear interferences on some time interval independent of the semiclassical parameter, and interferences for later time, thanks to explicit computations available for particular phases.

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非线性薛定谔方程的多相 WKB 分析中的非线性效应 *

我们考虑的是半经典极限下具有外部势能和立方非线性的薛定谔方程。初始数据是 WKB 状态之和，具有平滑相位和平滑、紧凑支撑的初始振幅，且具有不相交的支撑。我们证明，与线性情况一样，叠加原理在某个与半经典参数无关的时间间隔上成立，在初始数据的大小与半经典参数有关的几个条件下都是如此。当半自动参数的非线性效应较强时，我们引用可压缩欧拉方程的特性。对于较弱的非线性效应，我们证明在与半经典参数无关的某个时间间隔内可能不存在非线性干扰，而在之后的时间内则存在干扰，这要归功于对特定阶段的明确计算。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Nonlinearity 物理-物理：数学物理

CiteScore

3.00

自引率

5.90%

发文量

170

审稿时长

12 months

期刊介绍： Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest. Subject coverage: The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal. Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena. Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.