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

Abstract

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.

Geometric analysis of traveling wave solutions for the generalized KP-MEW-Burgers equation

“…The case of ω 1 ε 1 = 0, ω 2 ε 2 = 0 was studied in [17], and the results showed that system (20) presents periodic motion and chaotic behavior. For (n, a, b, r, µ, c, l, ω 1 , ω 2 , ε 1 , ε 2 )=(2, 0.045, 1, 0.08, 0.12, 0.4, 0.2, 1.25, 1.25, 0.5, 0) and the initial value (u, v) = (1, −0.42), system (20) presents periodic motion [17].…”

“…This phenomenon means system (20) presents periodic behavior. Let (n, b, r, µ, c, µ, c, l, ω 1 , ω 2 , ε 1 , ε 2 ) = (2, 1, 0.08, 0.12, 0.4, 0.2, 1.25, 2.5, 0.5, 0.5), and the initial values of systems (20) and ( 21) are (u, v) = (1, −0.42) and (u, v, w, z, p, q) = (1, −0.42, 0.8, 0.8, 0.6, 0.6), respectively. When a = 0.039, 0.046, 0.0534, zero is the largest Lyapunov exponent of system (21).…”

“…When a = 0.039, 0.046, 0.0534, zero is the largest Lyapunov exponent of system (21). System (20) presents 1-period, Let (n, b, r, µ, c, l, ω 1 , ω 2 , ε 1 , ε 2 ) = (2, 1, 0.08, 0.12, 0.4, 0.2, 1.25, √ 6, 0.5, 0.5), and the initial values of systems (20) and ( 21) are (u, v) = (1, −0.42) and (u, v, w, z, p, q) = (1, −0.42, 0.8, 0.8, 0.6, 0.6), respectively. In system (21), the largest Lyapunov exponent is zero when a = 0.035, 0.041.…”

“…In [22], Liu et al discuss static bifurcations in 2D differential systems based on the KCC geometric invariants and characterize the dynamics far from fixed points of the systems that generate bifurcations. [20] analyzes geometrically the traveling wave solution and chaos of the MEW-Burgers wave equation. To our knowledge, geometrization for the generalized KP-MEW-Burgers equation has not been reported.…”