The genesis of two-hump, W-shaped and M-shaped soliton propagations of the coupled Schr\"odinger-Boussinesq equations with conformable derivative

TL;DR

This paper investigates the propagation of two-hump, W-shaped, and M-shaped solitons in coupled Schr"odinger-Boussinesq equations with conformable derivatives, providing analytical solutions, boundedness proofs, and graphical illustrations.

Contribution

It introduces a method to find and analyze bounded soliton solutions in coupled equations with conformable derivatives, including integrability conditions and explicit solution forms.

Findings

Solutions maintain their shapes during propagation

Boundedness of solutions is rigorously proved

Graphical solutions illustrate soliton behaviors

Abstract

This work oversees with the coupled Schr\"odinger-Boussinesq equations with conformable derivative, which have lots of applications in laser and plasma. The said equations are reduced to a coupled stationary form using complex travelling wave transformation. Next Painlev\'e test applied to derived the integrable cases of the reduced equation, after that using RCAM derived the solution of reduced equations integrable and nonintegrable cases. Few theorems have been presented and proved to ensure their boundedness. All presented boundedness cases have been checked and explained by plotting the solutions for particulars values of parameters satisfying them. The obtained solutions of stationary form utilized to derive solutions of the coupled Schr\"odinger-Boussinesq equations with conformable derivative. The derived solutions have been plotted and explained. From this, it appears that these solutions propagate by maintaining their two-hump, W-shaped, M-shaped solutions shapes.