Exact dynamical behavior for a dual Kaup\u2013Boussinesq system by symmetry reduction and coupled trial equations method

Wen-He Li,Yong Wang

doi:10.1186/s13662-019-2357-0

Abstract

We propose a coupled trial equation method for a coupled differential equations system. Furthermore, according to the invariant property under the translation, we give the symmetry reduction of a dual Kaup–Boussinesq system, and then we use the proposed trial equation method to construct its exact solutions which describe its dynamical behavior. In particular, we get a cosine function solution with a constant propagation velocity, which shows an important periodic behavior of the system.

Highlights

The usual Kaup–Boussinesq system which reads⎧ ⎨ut = uxxx + 2(uv)x, ⎩vt = ux + vvx, describes the motion of water wave, where u(x, t) is the height of the water surface above a horizontal bottom and v(x, t) is the horizontal velocity
By proposing a coupled trial equation method and using symmetry reduction and a complete discrimination system for polynomial, we obtain its exact solutions which describe the dynamical behavior of the system
2 Trial equation method for a coupled system We propose a generalization of Liu’s trial equation method [9–14] to coupled differential equations systems as follows

Summary

Introduction

We can use the complete discrimination system for polynomial method to classify exact solutions for some nonlinear differential equations [17–25]. These methods have been extensively developed and applied to a lot of nonlinear problems [26–37]. By proposing a coupled trial equation method and using symmetry reduction and a complete discrimination system for polynomial, we obtain its exact solutions which describe the dynamical behavior of the system. We give the application of the proposed trial equation method to a dual Kaup–Boussinesq system

Symmetry and reduction

Conclusion