广义b方程的孤立波解及周期波解

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 670-686.

广义b方程的孤立波解及周期波解

杨佼朋¹(),梁勇^2,^*()

1.广东外语外贸大学数学与统计学院广州 510006
2.华南理工大学数学学院广州 510640

收稿日期:2023-11-21 修回日期:2024-01-04 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *梁勇，E-mail：dyliang@scut.edu.cn
作者简介:杨佼朋，E-mail：jpyang@oamail.gdufs.edu.cn
基金资助:
国家自然科学基金(12201136);广东省青年创新人才项目(2021KQNCX019);广州市基础与应用基础研究项目(202201010278)

Solitary and Periodic Solutions of the Generalized b-equation

Yang Jiaopeng¹(),Liang Yong^2,^*()

1. School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006
2. School of Mathematics, South China University of Technology, Guangzhou 510640

Received:2023-11-21 Revised:2024-01-04 Online:2024-06-26 Published:2024-05-17
Supported by:
NSFC(12201136);Foundation for Innovative Young Talents of GuangDong(2021KQNCX019);GuangZhou Basic and Applied Basic Research Foundation(202201010278)

摘要/Abstract

摘要：

对于广义 $b$ 方程的研究主要集中在 $b\ge0$ 的情况, 该文利用分支方法研究了 $b=-3$ 这类特殊广义 $b$ 方程的分支、非线性波解及动力学特征. 在一定参数条件下, 得到了该方程的分支相图, 还发现了不同于 $b>0$ 情况的新现象, 在行波系统中有无限多周期轨穿过奇异直线 $\varphi=c$. 同时, 给出了光滑孤立波解和光滑周期波解的存在性及其精确表达式, 共获得了 15 个非线性波解的显式表达式.

关键词: 广义 $b$ 方程, 定性理论, 分支方法, 孤立波解, 周期波解

Abstract:

The research of generalized $b$-equation mainly focuses on the case of $b\ge0$. This paper uses the bifurcation method to investigate the bifurcation, nonlinear wave solutions and dynamical characteristics of generalized $b$-equation with $b=-3$. Under certain parameter conditions, one obtains the bifurcation phase diagram of the equation. Meanwhile, a new phenomenon is found different from the case of $b>0$, in which an infinite number of periodic trajectories in the traveling wave system cross the singular line $\varphi=c$. The existence of solitary and periodic solutions is given, and the 15 exact expressions for nonlinear wave solutions are obtained.

Key words: Generalized $b$-equations, Qualitative theory, Bifurcation method, Solitary solutions, Periodic solutions

中图分类号:

O175.2

杨佼朋, 梁勇. 广义b方程的孤立波解及周期波解[J]. 数学物理学报, 2024, 44(3): 670-686.

Yang Jiaopeng, Liang Yong. Solitary and Periodic Solutions of the Generalized b-equation[J]. Acta mathematica scientia,Series A, 2024, 44(3): 670-686.

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