Investigating the irrotational gravity waves along the surface of an inviscid incompressible fluid model by modified nonlinear Schrödinger equation

This article discusses the solitons as well as additional solutions to the modified nonlinear Schrödinger equation, which appears in the study of optical waveguide and rogue waves in ocean. One theorem is analyzed and and also is investigated by obtaining one solution as exact optical soliton solution from the modified NLSE utilizing the Paul-Painlevé technique. Accurate traveling wave solutions are obtained using the generalized G-expansion equation method and the \(\tan (\phi /2)\) $tan(\varphi /2)$ -expansion method. In addition, semi-inverse variational principle method is used to finding the periodic wave, soliton and breather wave solution to mentioned nonlinear model. Restraint conditions for the occurrence of soliton and periodic wave solutions are also established. By selecting suitable values for the parameters involved in the solutions, 3D graphs, 2D density graphs and line graphs are presented to provide graphical demonstration of the results. The reported results will be helpful to design new and better optical devices. The findings of this research may also help in problems arising in ocean engineering or nonlinear optics.