非线性光学晶格中的梯度流方法

数学物理学报 ›› 2019, Vol. 39 ›› Issue (5): 1170-1182.

非线性光学晶格中的梯度流方法

张瑞凤^1,*(),刘男²()

¹ 河南大学数学与统计学院河南开封 475004
² 北京应用物理与计算数学研究所北京 100088

收稿日期:2018-05-04 出版日期:2019-10-26 发布日期:2019-11-08
通讯作者: 张瑞凤 E-mail:zrf615@henu.edu.cn;ln10475@163.com
作者简介:刘男, E-mail:ln10475@163.com
基金资助:
国家自然科学基金(11471099);国家自然科学基金(11671120)

Gradient Flow Method in Nonlinear Optical Lattices

Ruifeng Zhang^1,*(),Nan Liu²()

¹ College of Mathematics and Statistics, Henan University, Henan Kaifeng 475004
² Institute of Applied Physics and Computational Mathematics, Beijing 100088

Received:2018-05-04 Online:2019-10-26 Published:2019-11-08
Contact: Ruifeng Zhang E-mail:zrf615@henu.edu.cn;ln10475@163.com
Supported by:
the NSFC(11471099);the NSFC(11671120)

摘要/Abstract

摘要：

该文利用梯度流方法研究非线性光学晶格中经典薛定谔方程稳态解的存在性.文中首先给出了控制方程整体解的存在性，然后证明了当时间趋于无穷大时整体解收敛到一个平衡态（即光学晶格模型的稳态解）.此外，通过Łojasiewicz-Simon不等式给出了收敛速度估计.

关键词: 梯度流方法, 薛定谔方程, 稳态解, Łojasiewicz-Simon不等式

Abstract:

In this paper, we study the existence of the steady state solutions for a classical Schrödinger equation in nonlinear optical lattices by means of gradient flow method. We first establish the existence of a global solution of the governing parabolic equation. Then we prove the convergence of the global solution to an equilibrium (i.e., a steady state solution in optical lattices model) as time goes to infinity. Furthermore, we provide an estimate on the convergence rate by using the Łojasiewicz-Simon inequality.

Key words: Gradient flow method, Schrödinger Equation, Łojasiewicz-Simon inequality, Steady state solutions

中图分类号:

O175.26

张瑞凤,刘男. 非线性光学晶格中的梯度流方法[J]. 数学物理学报, 2019, 39(5): 1170-1182.

Ruifeng Zhang,Nan Liu. Gradient Flow Method in Nonlinear Optical Lattices[J]. Acta mathematica scientia,Series A, 2019, 39(5): 1170-1182.

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