On soliton solutions of a modified nonlinear Schrödinger's equation of third-order governing in optical fibers

Recently, the development of novel methods for finding solutions to partial differential equations has made great progress in various fields of science. These equations are an important tool in describing many phenomena that occur around us. The research work aims to study a third-order nonlinear Schrödinger's equation using the modified generalized exponential rational function method, and the extended sinh-Gordon equation expansion method. The common idea between the two methods is that it is first necessary to reduce the equation with partial derivatives to a form of the equation with ordinary derivatives using a new wave definition. Moreover, the retrieved solutions are obtained in a closed form of elementary functions which are important in practical applications. We successfully construct some soliton, singular soliton and singular periodic wave solutions to the nonlinear complex model. Further, numerical simulations are also included via several figures. A significant advantage of both techniques is the ability to introduce a wide range of types of analytical solutions in terms of well-known elementary functions. It is obtained that these methods are also applicable in solving other equations in nonlinear science.