Bifurcation of the travelling wave solutions in a perturbed (1+1)-dimensional dispersive long wave equation via a geometric approach

被引:2
作者
Zheng, Hang [1 ]
Xia, Yonghui [2 ]
机构
[1] Wuyi Univ, Dept Math & Comp, Wuyishan 354300, Peoples R China
[2] Foshan Univ, Sch Math & Big Data, Foshan 528000, Peoples R China
基金
中国国家自然科学基金;
关键词
wave equation; solitary wave solutions; geometric singular perturbation; RATIONAL EXPANSION METHOD; KELLER-SEGEL SYSTEM; PERIODIC-WAVES; HOMOCLINIC ORBITS; SOLITARY WAVES; EXISTENCE; DIFFUSION; KINK;
D O I
10.1017/prm.2024.45
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Choosing ${\kappa }$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$-dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${\kappa }\in \left (0,1-\frac {\sqrt 3}{3}\right )\cup \left (1+\frac {\sqrt 3}{3},2\right )$, while the kink and anti-kink wave solutions exist at a unique wave speed $c<^>*=\sqrt {15}/3$ for $\kappa =0$ or $\kappa =2$. The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.
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页数:28
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