A finite difference scheme for (2+1)D cubic-quintic nonlinear Schrödinger equations with nonlinear damping

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED Applied Numerical Mathematics Pub Date : 2024-07-24 DOI:10.1016/j.apnum.2024.07.008

Anh Ha Le , Toan T. Huynh , Quan M. Nguyen

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Abstract

Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of $n \geq 2$ suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete $L^{2}$ -norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete $L^{2}$ -norm and $H^{1}$ -norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence.

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具有非线性阻尼的 (2+1)D 立方五次方非线性薛定谔方程的有限差分方案

空间维数 n≥2 的纯立方非线性薛定谔方程的孤子会发生临界和超临界坍缩。这些孤子可以在立方五次元非线性介质中稳定下来。本文分析了具有立方阻尼的 (2+1)D 立方-五次方非线性薛定谔方程的 Crank-Nicolson 有限差分方案。我们证明，离散 L2 值的离散解和离散能量都是有界的。通过使用适当的设置和估计，证明了数值解的存在性和唯一性。此外，还建立了离散 L2 规范和 H1 规范下空间和时间的二阶误差估计。对具有立方阻尼的 (2+1)D 立方五次方非线性薛定谔方程进行了数值模拟，以验证其收敛性。

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来源期刊

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.

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