Abstract
This paper considers issues such as integrability and how to get specific classes of solutions for nonlinear differential equations. The nonlinear Kundu–Mukherjee–Naskar (KMN) equation is chosen as a model, and its traveling wave solutions are investigated by using a direct solving method. It is a quite recent proposed approach called the functional expansion and it is based on the use of auxiliary equations. The main objectives are to provide arguments that the functional expansion offers more general solutions, and to point out how these solutions depend on the choice of the auxiliary equation. To see that, two different equations are considered, one first order and one second order differential equations. A large variety of KMN solutions are generated, part of them listed for the first time. Comments and remarks on the dependence of these solutions on the solving method and on form of the auxiliary equation, are included.
Highlights
The nonlinear phenomena are as frequent in nature as the linear ones
This paper deals with a special method for solving nonlinear ordinary differential equation (NODE), the functional expansion, and it has a triple aim: (i) checking if and how the method is functioning for a specific nonlinear system, namely, for the Kundu–Mukherjee–Naskar (KMN) model; (ii) proving that the method leads to more general solutions as, for example, those given by G 0 /G approach; and (iii) discussing the sensitivity of the solving method on the choice of the auxiliary equation
The paper tackled the general problem of integrability of the nonlinear dynamical systems [33,34]
Summary
The nonlinear phenomena are as frequent in nature as the linear ones. They are usually described by nonlinear differential equations, so that solving nonlinear partial differential equations (NPDEs) is quite an important issue. This paper deals with a special method for solving NODEs, the functional expansion, and it has a triple aim: (i) checking if and how the method is functioning for a specific nonlinear system, namely, for the Kundu–Mukherjee–Naskar (KMN) model; (ii) proving that the method leads to more general solutions as, for example, those given by G 0 /G approach; and (iii) discussing the sensitivity of the solving method on the choice of the auxiliary equation. As explained, this strategy was adopted in order to conclude on how the solutions depend on the choice of the auxiliary equation.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
