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Some explicit travelling wave solutions to constructing exact solutions of nonlinear partial differential equations of mathematical physics are presented. By applying a theory of Frobenius decompositions and, more precisely, by using a transformation method to the coupled Burgers, combined Korteweg-de Vries- (KdV-) modified KdV and Schrödinger-KdV equation is written as bilinear ordinary differential equations and two solutions to describing nonlinear interaction of travelling waves are generated. The properties of the multiple travelling wave solutions are shown by some figures. All solutions are stable and have applications in physics.

The investigation of traveling wave solutions of nonlinear evolution equations (NLEEs) plays a vital role in different branches of mathematical physics, engineering sciences, and other technical arenas, such as plasma physics, nonlinear optics, solid state physics, fluid mechanics, chemical physics, and chemistry.

The Burgers’ equation has been found to describe various kinds of phenomena such as a mathematical model of turbulence [

The Korteweg-de Vries (KdV) equation which models shallow-water phenomena has been analyzed extensively using the invariance properties that occur from the Lie point symmetry generator that admits it. In particular, travelling wave solutions arise from the combination of translations in space and time. Also, Galilean invariants and scale-invariant solutions are dependent on first and second Painleve transcendent [

The topic of solitons produced by nonlinear interactions is a very fundamental topic in various fields, including optical solitons in fibers [

In recent years, various methods have been established to obtain exact traveling solutions of nonlinear partial differential equations, for example, the Jacobi elliptic function expansion method [

This paper is organized as follows. An introduction is given in Section

Now, we simply describe the generalized extended tanh-function method. Consider a given system of NEEs, say, in two variables,

There exist the following steps to be considered further.

Determine the

Substituting (

By solving the system, we may determine the above parameters.

Substituting the parameters

The coupled Burgers’ equations [

Balancing the nonlinear term

By the solution of the above system of equations, we can find

Then, combining (

The shape solutions (

Travelling waves solutions (

Consider the combined KdV-modified KdV equation:

Travelling waves solutions (

Consider the coupled Schrödinger-KdV equations (Davey-Stewartson)

Substituting from (

The shape solutions (

Travelling waves solutions (

The authors declare that there is no conflict of interests regarding the publication of this paper.