Abstract
In this paper we obtain exact solutions of the -dimensional higher-order Broer-Kaup system which was obtained from the Kadomtsev-Petviashvili equation by the symmetry constraints. The methods used to determine the exact solutions of the underlying system are the Lie group analysis and the simplest equation method. The solutions obtained are the solitary wave solutions. Moreover, we derive the conservation laws of the -dimensional higher-order Broer-Kaup system by employing the multiplier approach and the new conservation theorem.
Highlights
1 Introduction In this paper we study the ( + )-dimensional higher-order Broer-Kaup system ut + uxx + u – uux + uv x =, vt + vxx + u v + uvx + v x =, ( . a) ( . b) which was first introduced by Lou and Hu [ ] by considering the symmetry constraints of the Kadomtsev-Petviashvili equation
Ut – uxx + uux – vx =, vt + vxx –x =, ( . a) ( . b) which is used to model the bi-directional propagation of long waves in shallow water
It has proved to be a versatile tool for solving nonlinear problems described by the differential equations arising in mathematics, physics and in other scientific fields of study
Summary
In the latter half of the nineteenth century, Sophus Lie ( - ) developed one of the most powerful methods to determine solutions of differential equations. We use the Lie group analysis approach along with the simplest equation method to obtain exact solutions of the ( + )-dimensional higher-order Broer-Kaup system X The symmetry X gives rise to the group-invariant solution of the form u = F(z), v = G(z), where z = x is an invariant of X and the functions F and G satisfy the following system of ODEs:. X By solving the corresponding Lagrange system for the symmetry X , one obtains an invariant z = xt– / and the group-invariant solution of the form u = t– / F(z), v = t– / G(z), where the functions F and G satisfy the following system of ODEs:. It is well known that the Bernoulli and Riccati equations are nonlinear ODEs whose solutions can be written in terms of elementary functions
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