New approximate-analytical solutions for the nonlinear fractional Schr\"{o}dinger equation with second-order spatio-temporal dispersion via double Laplace transform method

TL;DR

This paper develops a new analytical approach combining double Laplace transform and Adomian decomposition to solve a fractional nonlinear Schrödinger equation with spatio-temporal dispersion, providing approximate solutions in different fractional senses.

Contribution

It introduces a generalized double Laplace transform method coupled with Adomian decomposition for solving fractional nonlinear Schrödinger equations with dispersion.

Findings

Derived approximate analytical solutions in Caputo and conformable derivatives.

Compared solutions graphically to analyze differences.

Extended the method to a nonlinear fractional Schrödinger equation.

Abstract

In this paper, a modified nonlinear Schr\"{o}dinger equation with spatio-temporal dispersion is formulated in the senses of Caputo fractional derivative and conformable derivative. A new generalized double Laplace transform coupled with Adomian decomposition method has been defined and applied to solve the newly formulated nonlinear Schr\"{o}dinger equation with spatio-temporal dispersion. The approximate analytical solutions using the proposed generalized method in the sense of Caputo fractional derivative and conformable derivatives are obtained and compared with each other graphically.