On the exact and numerical complex travelling wave solution to the nonlinear Schrödinger equation

Abstract

In this article, the Nonlinear Schrödinger (NLS) equation, which is very important in physics, have been discussed. The analytical solution of this equation with the $(1/G^{\prime })$-expansion method and its numerical solutions with the Finite Difference Method have been presented. $(1/G^{\prime })$-expansion method has been successfully applied to NLS equation and different type complex hyperbolic travelling wave solutions have been produced. An initial condition for the NLS equation has been created using this solution. This solution has been used to create the initial condition required for FDM. Truncation Error, Stability analysis and $L_2$ and $L_{\mathrm{\infty }}$ norm errors of the numerical results obtained have been examined. Besides, FDM has been successfully applied to NLS equation and numerical solutions have been produced. Besides, both analytical and numerical solutions are supported with graphics and tables. The results obtained by this numerical method have been compared with the exact solution.

Keywords:

• Nonlinear Schrödinger equation
• $(1/G^{\prime })$-expansion method
• finite difference method
• complex hyperbolic travelling wave solutions

AMS Classification 2010:

• 65L12
• 74S20

Disclosure statement

No potential conflict of interest was reported by the author(s).