Analysis and simulation of periodic and solitary waves in nonlinear dispersive-dissipative solids
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.
scientific article; zbMATH DE number 7382119
scientific article