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

This work oversees with the coupled Schrödinger-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 Painlevé 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 Schrödinger-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.