Qualitative geometric analysis of traveling wave solutions of the modified equal width Burgers equation

This paper devotes to the qualitative geometric analysis of the traveling wave solutions of MEW-Burgers wave equation. Firstly, MEW-Burgers equation is transformed into an equivalent planar dynamical system by using traveling wave transformation. Then the global structure of the planar system is presented, and solitary waves, kink waves (anti-kink waves), and periodic waves are found. Secondly, Jacobi stability for the planar system is studied based on KCC theory, and Jacobi stability of any point on the trajectory of system is dicussed. The dynamical behavior of the deviation vector near the equilibrium points is analyzed, and the numerical simulation is consistent with the theoretical analysis. Lyapunov stability and Jacobi stability of the equilibrium points are also compared and analyzed. The obtaining results show that Lyapunov stability of the equilibrium points of the system is not exactly consistent with Jacobi stability. Finally, the planar system with periodic disturbance is transformed into a six dimensional nonlinear system, and the periodic, quasi-periodic, and chaotic dynamical behaviors of the system are numerically simulated.

Automated summary

This paper focuses on the qualitative geometric analysis of traveling wave solutions of the MEW-Burgers wave equation. It transforms the MEW-Burgers equation into an equivalent planar dynamical system using the traveling wave transformation. The global structure of the planar system is presented, and solitary waves, kink waves, and periodic waves are found. The paper then studies the Jacobi stability of the planar system based on KCC theory, analyzing the dynamical behavior near equilibrium points and comparing Lyapunov stability and Jacobi stability. It also transforms the planar system with periodic disturbance into a six-dimensional nonlinear system and numerically simulates the periodic, quasi-periodic, and chaotic dynamical behaviors of the system.