Symbolic computation to construct new soliton solutions and dynamical behaviors of various wave structures for two different extended and generalized nonlinear Schrödinger equations using the new improved modified generalized sub-ODE proposed method

In this work, we propose a new improved modified generalized sub-ODE method for constructing new solitons and traveling wave solutions, and also show the dynamical behaviors of various wave structures to the extended nonlinear Schrödinger equation with higher-order odd and even terms, as well as a generalized nonlinear Schrödinger equation of fourth-order, using symbolic computerized work via Mathematica. This newly proposed method improves and modifies the Li method (Li-Hua and Jin-Yu, 2009) The improved method presented in this paper can be used to solve other nonlinear equations with nonlinear terms of any order of physical systems in order to obtain many solitary wave solutions and other traveling wave solutions for such nonlinear models in a unified manner. The resulting soliton solutions and other forms of solutions are very useful and advantageous in many branches of mathematical physics and nonlinear sciences such as ocean engineering, optical fibers, plasma physics, and fluid dynamics.