Qualitative analysis to traveling wave solutions of Zakharov–Kuznetsov–Burgers equation and its damped oscillatory solutions

Abstract

The theory of planar dynamical systems is applied in this paper to carry out a qualitative analysis to the planar dynamical system corresponding to the bounded traveling wave solution of the Zakharov–Kuznetsov–Burgers equation, and obtain the existence and uniqueness of the bounded traveling wave solutions. According to the discussions on the relationships between the shapes of bounded traveling wave solutions and the dissipation coefficient d, a critical value d0 is found for arbitrary traveling wave speed v and integral constant g. This equation has a unique monotone kink profile solitary wave solution as the dissipation coefficient d satisfies |d¯|>d0; while it has a unique damped oscillatory solution as |d¯|<d0. This paper also presents the exact bell profile solitary wave solution as d=0. Furthermore, we appropriately design the structure of the damped oscillatory solution in light of the evolution relationships of the solution orbit in the global phase portraits to which the damped oscillatory solution corresponds and obtain its approximate solution by means of the undetermined coefficients method. Finally, based on the integral equation that reflects the relationships between the approximate damped oscillatory solution and its exact solution, the error estimate is given for the approximate damped oscillatory solution. The error is infinitesimal decreasing in exponential form.