On novel analytical solutions to a generalized Schrödinger’s equation using a logarithmic transformation-based approach

Abstract

Investigating analytical solutions of nonlinear Schrd̈inger equations in the form of solitary waves can be quite a challenge, but it holds the promise of uncovering fascinating discoveries. In this paper we discover novel solitary wave solutions to the Schrödinger equation, taking into account important aspects such as dispersion, diffraction, and Kerr nonlinearity. To achieve this goal, we utilized an efficient technique that involved applying logarithm transformations. This approach effectively converts the original equation into an ordinary differential equation, presenting symbolic solutions. The resulting solutions exhibit diverse forms such as trigonometric, hyperbolic, and rational functions, finding applications in various fields including plasma physics, nonlinear optics, optical fibers, and nonlinear sciences. The presence of selectively chosen constants in these solutions results in rich and complex dynamical behavior. Furthermore, the article showcases several numerical simulations that correspond to these findings. An important aspect to note is that the methodologies employed in this research have not been previously utilized to solve the model under investigation. Additionally, these techniques can be easily adapted to address a diverse range of other nonlinear problems.