Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method

Hossein Aminikhah,Hadi Rezazadeh,Amir Hosein Refahi Sheikhani

doi:10.5269/bspm.v34i2.25501

Abstract

In this paper, we will use the functional variable method to construct exact solutions of some nonlinear systems of partial differential equations, including, the (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation, the WhithamBroer-Kaup-Like systems and the Kaup-Boussinesq system. This approach can also be applied to other nonlinear systems of partial differential equations which can be converted to a second-order ordinary differential equation through the travelling wave transformation.

Highlights

The functional variable methodWe describe the main steps of the functional variable method for finding exact solutions of nonlinear system of partial differential equations
In this paper, we will use the functional variable method to construct exact solutions of some nonlinear systems of partial differential equations, including, the (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation, the WhithamBroer-Kaup-Like systems and the Kaup-Boussinesq system
The advantage of this method is that one treats nonlinear problems by essentially linear methods, based on which it is easy to construct in full the exact solutions such as soliton-like waves, compacton and noncompacton solutions, trigonometric function solutions, pattern soliton solutions, black solitons or kink solutions, and so on

Summary

The functional variable method

We describe the main steps of the functional variable method for finding exact solutions of nonlinear system of partial differential equations. Applying the travelling wave transformations u(x, t) = U (ξ) and v(x, t) = V (ξ) where ξ = x − wt, converts Eq(2.1) into a system of ordinary differential like. The system (2.2) is converted into a secondorder ordinary differential equation as. Travelling wave solutions of nonlinear systems of PDEs by using the . Substituting (2.5) into Eq(2.3) and after the mathematical manipulations, we reduce the ordinary differential equation (2.3) in terms of U , F as. The key idea of this particular form Eq(2.6) is of special interest because it admits analytical solutions for a large class of nonlinear wave type equations.

Applications

The Whitham-Broer-Kaup-Like systems

The Kaup-Boussinesq system

Conclusion