BIFURCATIONS, SMOOTH AND NON-SMOOTH TRAVELING WAVE SOLUTIONS FOR GENERALIZED NONLINEAR DISPERSIVE BOUSSINESQ EQUATION
Keywords:
generalized nonlinear dispersive Boussinesq equation, bifurcation, phase portrait, soliton cusp wave, periodic cusp waveDOI:
https://doi.org/10.17654/0972111824007Abstract
The traveling wave equation of the generalized nonlinear dispersive Boussinesq equation is a singular system with two possible singularities, which is transformed into a regular system after making a scale transformation. The bifurcations and phase portraits of the regular system are obtained by using bifurcation theory of dynamical systems. The singular traveling wave theory is used to discuss the impact of singularity on the smoothness of solutions and analyze the reason for the appearance of non-smooth solutions such as soliton cusp waves and periodic cusp waves. Under different parameter conditions, the sufficient conditions for the existence of smooth and non-smooth traveling wave solutions of the singular system are given. For certain special cases, the parametric representations of 25 explicit and exact traveling wave solutions are presented such as smooth peak-shaped, valley-shaped, and kink-shaped soliton waves along with non-smooth soliton cusp waves. Furthermore, 2D wave plots of smooth soliton waves and non-smooth soliton cusp wave solutions are drawn to visualize the dynamics of the equation. The results of this study contribute to the better understanding of the generalized nonlinear dispersive Boussinesq equation.
Received: April 10, 2024
Accepted: June 1, 2024
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