Analysis and simulation of periodic and solitary waves in nonlinear dispersive-dissipative solids
type: Item
instance of: scholarly article
title: Analysis and simulation of periodic and solitary waves in nonlinear dispersive-dissipative solids
zbMATH Open document ID: 1479.34078
author: Yanni Zeng
author: Xianbo Sun
author: P. Yu
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2021-08-16T00:00:00Z
review text: The paper is concerned with the nonlinear elastic rod model embedded in an external medium with weak disturbance \[ v_{tt}-\alpha_1v_{xx}-\alpha_3(v^2)_{xx}-\alpha_4 v_{xxxx}+\alpha_5 v_{xxtt}=\epsilon (\beta_2 v_{xxt}+\beta_6 (v^2)_{xxt}+\beta_7 v_{xxxxt}-\beta_8 v_{xxttt}) \tag{1} \] with \(0<\epsilon \ll 1\), and the coefficients are real numbers. By using the geometric singular perturbation theory, the classical Bogdanov-Takens bifurcation theory, and the Chebyshev criterion, the authors prove the existence of a unique periodic wave of equation (1). Under some assumptions on the parameters, they show that this periodic wave appears near a constant stationary state, and then it amplifies and degenerates into a solitary wave. Some numerical examples with real data which illustrate the analytic predictions are also presented.
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 35C07
zbMATH DE Number: 7382119
zbMATH Keywords: nonlinear dispersive-dissipative solids
zbMATH Keywords: periodic traveling wave
zbMATH Keywords: Melnikov function
zbMATH Keywords: homoclinic cycle
zbMATH Keywords: geometric singular perturbation theory
zbMATH Keywords: Chebyshev criterion
reviewed by: Rodica Luca
MaRDI profile type: Publication
full work available at URL: j.cnsns.2021.105921
OpenAlex ID: W3169595361
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DOI: 10.1016/J.CNSNS.2021.105921
description: scientific article; zbMATH DE number 7382119