Existence of periodic waves for a perturbed quintic BBM equation

Lina Guo; Yulin Zhao

doi:10.3934/dcds.2020198

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2020, Volume 40, Issue 8: 4689-4703. Doi: 10.3934/dcds.2020198

This issue Previous Article Fluctuations of ergodic sums on periodic orbits under specification Next Article Representation formula for symmetrical symplectic capacity and applications

Existence of periodic waves for a perturbed quintic BBM equation

Lina Guo and
Yulin Zhao^,

School of Mathematics(Zhuhai), Sun Yat-Sen university, Zhuhai 519082, China

^* Corresponding author: Yulin Zhao
^* Corresponding author: Yulin Zhao

Received: April 30, 2019

Revised: January 31, 2020

Published: May 2020

This research is supported by the NSF of China (No.11971495 and No.11801582)

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Abstract

This paper dealt with the existence of periodic waves for a perturbed quintic BBM equation by using geometric singular perturbation theory. By analyzing the perturbations of the Hamiltonian vector field with a hyperelliptic Hamiltonian of degree six, we proved that periodic wave solutions persist for sufficiently small perturbation parameter. It is also proved that the wave speed $ c_0(h) $ is decreasing on $ h $ by analyzing the ratio of Abelian integrals, where $ h $ is the energy level value. Moreover, the upper and lower bounds of the limit wave speed are given.

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Mathematics Subject Classification: Primary: 34C25, 34C60; Secondary: 37C27.

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Figure 1. The phase portrait of system (1.13)

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