Global Existence and Uniqueness of Periodic Waves for a Perturbed Combined&nbsp;Double-Dispersive Equation

LIAO Xiao-zhao, HUANG Wen-tao, YANG Su-min

doi:10.13371/j.cnki.chin.q.j.m.2022.02.002

Chinese Quarterly Journal of Mathematics >

2022 , Vol. 37 >Issue 2: 124 - 131

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2022.02.002

Global Existence and Uniqueness of Periodic Waves for a Perturbed Combined Double-Dispersive Equation

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1. School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China;
2. School of Public Administration, Guangxi Technological College of Machinery and Electrcity,
Nanning 530007, China

LIAO Xiao-zhao (1996-), male, native of Yueyang, Hunan, postgraduate of Guangxi Normal University, engages in ordinary differential equation; HUANG Wen-tao (1966-), male, native of Guilin, Guangxi, professor of Guangxi Normal University, engages in ordinary differential equation; YANG Su-min (1984-), female, native of Fugou, Henan, lecturer of Guangxi Technological College of Machinery and Electrcity, engages in dynamical systems.

Received date: 2021-12-04

Online published: 2022-06-30

Supported by

Supported by the National Natural Science Foundation of China (Grant No. 12061016) and the Applied Mathematics Center of Guangxi; Foundation of Guangxi Technological College of Machinery and Electrcity (Grant No. 2021YKYZ010).

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Abstract

In this paper, we study the periodic wave propagation phenomenon in elastic waveguides modeled by a combined double-dispersive partial differential equation (PDE). The traveling wave ansazt transforms the PDE model into a perturbed integrable ordinary differential equation (ODE). The global bifurcation theory is applied for the perturbed ODE model to establish the existence and uniqueness of the limit cycle, which corresponds
the periodic traveling wave for the PDE model. The main tool is the Abelian integral taken from Poincar´ e bifurcation theory. Simulation is carried out to verify the theoretical result.

Key words： Combined double-dispersive PDE; Abelian integral; Limit cycle; Hamiltonian function; Periodic wave

Cite this article

LIAO Xiao-zhao, HUANG Wen-tao, YANG Su-min . Global Existence and Uniqueness of Periodic Waves for a Perturbed Combined Double-Dispersive Equation[J]. Chinese Quarterly Journal of Mathematics, 2022 , 37(2) : 124 -131 . DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.002

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