# Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation

## DOI:

https://doi.org/10.53391/mmnsa.2021.01.003## Keywords:

Eckhaus equation, cubic nonlinear Klein Gordon equation, complex hyperbolic trigonometric travelling wave solutions, non-integer balancing term## Abstract

In this study, an alternative method has been applied to obtain the new wave solution of mathematical equations used in physics, engineering, and many applied sciences. We argue that this method can be used for some special nonlinear partial differential equations (NPDEs) in which the balancing methods are not integer. A number of new complex hyperbolic trigonometric traveling wave solutions have been successfully generated in the Eckhaus equation (EE) and nonlinear Klein-Gordon (nKG) equation models associated with the Schrödinger equation. The graphs representing the stationary wave are presented by giving specific values to the parameters contained in these solutions. Finally, some discussions about new complex solutions are given. It is discussed by giving physical meaning to the constants in traveling wave solutions, which are physically important as well as mathematically. These discussions are supported by three-dimensional simulation. In order to eliminate the complexity of the process and to save time, computer package programs have been utilized.

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*Mathematical Modelling and Numerical Simulation With Applications (MMNSA)*,

*1*(1), 24–31. https://doi.org/10.53391/mmnsa.2021.01.003

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