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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., Comment: 19 pages 5 figures

In this article, a modification of the rapidly convergent approximation method is proposed to solve a coupled Korteweg-de Vries equations with conformable derivative that govern shallow-water waves. Based on the Leibniz and chain rule of conformable derivative, these equations reduced into ODEs with integer-order using traveling wave transformation. Adopting the modified scheme a new novel exact solution of the reduced coupled ordinary differential equations is obtained in terms of exponential functions. Finally, by putting them back into traveling wave transformation the solutions of the considered partial differential equations with conformable derivative are derived. To ensure the boundedness of the derived solutions few theorems have been proposed and proved. The derived results of the theorems are utilized to plot the solutions. Graphics exhibit that solutions have variant multi-hump soliton peculiarities and their tails decay to zero exponentially in a monotonic manner. These results not only show the efficiency of the modified scheme but also establish that the solution is enriched with new multi-hump features.

In this work, an efficient approximation scheme has been proposed for getting accurate approximate solution of nonlinear partial differential equations with constant or variable coefficients satisfying initial conditions in a series of exponential instead of an algebraic function of independent variables. As a consequence: i) the convergence of the series found to be faster than the same obtained by few other methods and ii) the exact analytic solution can be obtained from the first few terms of the series of the approximate solution, in cases the equation is integrable. The convergence of the sum of the successive correction terms has been established and an estimate of the error in the approximation has also been presented. The efficiency of the present method has been illustrated through some examples with a variety of nonlinear terms present in the equation., Comment: 19 pages, 2 figures