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

• The uniqueness of the bounded traveling wave solution is proved for ZKB equation.
• We give a critical value d0d0 characterizing the scale of dissipation effect.
• We obtain the approximate damped oscillatory solution.
• We present the error estimate between approximate solution and its exact solution.
• The method in this paper can also be applied to KdV–Burgers equation.

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 d0d0 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¯|