Solving the Nonlinear Cubic Schrödinger Equation by an Extended Complex Tanh-Function Approach

Abstract

The nonlinear cubic Schrödinger equation not only starts from realistically physical phenomena, but can also be widely used to many physically significant areas such as fluid dynamics, condensed matter physics, plasma physics, quantum mechanics, nonlinear optics and superconductivity. As a consequence, it is a very significant and challenging topic to research the explicit and accurate travelling wave solutions to the nonlinear cubic Schrödinger equation. In this work, based on the ideas of the complex tanh-function method and the extended tanh-function method, an extended complex tanh-function approach is presented for constructing the explicit and accurate travelling wave solutions of nonlinear Schrödinger-type equations. Crucial to our technique is to take full advantage of a complex Riccti equation containing a parameter b and to employ its solutions to replace the tanh function in the complex tanh-function method. It is quite interesting that the sign of the parameter b can be applied to exactly judge the numbers and types of traveling wave solutions. We have illustrated its feasibility by application to the nonlinear cubic Schrödinger equation. As a result, some explicit and accurate travelling wave solutions of the nonlinear cubic Schrödinger equation are successfully investigated in a simple manner. Our approach can not only obtain the all solutions given in Ref [21], but also derive solutions that cannot be seen in Ref [21]. In addition, compared with the proposed approaches in the existing references, the approach described herein appears to be less calculative. Our technique may provide a novel way of thinking for solving nonlinear Schrödinger-type equations. We believe that the procedure used herein may also be applied to explore the explicit and accurate travelling wave solutions of other nonlinear Schrödinger-type equations. We try to generalize this approach to search for the explicit and accurate travelling wave solutions of other ordinary coefficient even variable coefficient nonlinear Schrödinger-type equations.