﻿ 具有分布时滞的奇异扰动KdV-KS方程的孤立波 Solitary Waves of Singularly Perturbed KdV-KS Equation with Distributed Delay

Pure Mathematics
Vol.04 No.06(2014), Article ID:14391,9 pages
10.12677/PM.2014.46037

Solitary Waves of Singularly Perturbed KdV-KS Equation with Distributed Delay

Yongxin Jiang

College of Science, Hohai University, Nanjing

Email: yxinjiang@126.com

Received: Oct. 13th, 2014; revised: Nov. 10th, 2014; accepted: Nov. 14th, 2014

Copyright © 2014 by author and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

ABSTRACT

We study a sort of nonlinear reaction diffusion equation based on the Korteweg-de Vries (KdV) equation with a convolution term which introduces a time delay in the nonlinearity and with a higher order singularly perturbing term as the Kuramoto-Sivashinsky (KS) equation, called KdV-KS equation. We focus on the question of the existence of solitary wave solutions. By using geometric singular perturbation analysis and the linear chain trick, we prove that the solitary wave solutions persist when the average delay is suitably small. The explicit expression of original KdV solitary is not required.

Keywords:Geometric Singular Perturbation, KdV Equation, Solitary Wave, Distributed Delay

Email: yxinjiang@126.com

1. 引言

Korteweg de Vries方程(简称KdV方程)

(1)

(2)

Ercolani [4] 研究了其周期解，T. Ogawa [5] 研究了其同宿轨和孤立波，范兴华等 [6] 分析了具有更复杂非线性项的mKdV-KS方程，用的主要方法是几何奇异摄动理论 [7] [8] 。这种方法也用来分析其他的反映扩散方程 [9] [10] 。和上面提到的传统方法相比，这种方法不需要原始KdV方程的显式解的表达式。几何奇异摄动方法对给出扰动解的第一个图形起着非常特殊的作用，它发展了一种非常具有洞察力的思想，它可以在相对简单的极限形式附近构造非常复杂的动力学行为。一些著名的现象比如松弛振子，鸭解等都能通过几何奇异摄动方法来得到。

(3)

(3)的孤立波是一种特殊的行波解，其中是速度，且满足下面的泛函微分方程：

(4)

. (5)

(6)

2. 预备知识

(7)

(I) 位于邻域内，且同胚于；并且，它在(7)的流下是局部不变的且对的。

(II) 对某个函数

(III) 存在局部不变稳定和不稳定流形它们与同胚且位于其邻域中；并且，它在(7)的流下是局部不变的且对的。

, (8)

.

, (9)

(10)

3. 孤立波的存在性

. (11)

. (12)

,

. (13)

(14)

(15)

.

. (16)

. (17)

(18)

.

, (19)

(20)

(21)

(22)

(23)

(24)

. (25)

(26)

.

，我们得到

.

.

Solitary Waves of Singularly Perturbed KdV-KS Equation with Distributed Delay. 理论数学,06,251-260. doi: 10.12677/PM.2014.46037

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