Shape analysis of bounded traveling wave solutions and solution to the generalized Whitham-Broer-Kaup equation with dissipation terms

Abstract

This paper deals with the problem of the bounded traveling wave solutions’ shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for short. The authors employ the theory and method of planar dynamical systems to make comprehensive qualitative analyses to the above equation satisfied by the horizontal velocity component u(ξ) in the traveling wave solution (u(ξ), H(ξ)), and then give its global phase portraits. The authors obtain the existent conditions and the number of the solutions by using the relations between the components u(ξ) and H(ξ) in the solutions. The authors study the dissipation effect on the solutions, find out a critical value r*, and prove that the traveling wave solution (u(ξ),H(ξ)) appears as a kink profile solitary wave if the dissipation effect is greater, i.e., |r| ≥ r*, while it appears as a damped oscillatory wave if the dissipation effect is smaller, i.e., |r| < r*. Two solitary wave solutions to the WBK equation without dissipation effect is also obtained. Based on the above discussion and according to the evolution relations of orbits corresponding to the component u(ξ) in the global phase portraits, the authors obtain all approximate damped oscillatory solutions (ũ(ξ), \(\tilde H\)(ξ)) under various conditions by using the undetermined coefficients method. Finally, the error between the approximate damped oscillatory solution and the exact solution is an infinitesimal decreasing exponentially.