Abstract
In this article, the non-linear coupled Hirota equations are considered attributed to localized excitations. By employing a modified double Laplace transform decomposition method, we derived the general approximate solutions for the coupled Hirota and coupled Hirota Satsuma equations. It is observed that the proposed method is a systematic and powerful scheme for solving such nonlinear systems. The main advantage of the method is to calculate solutions in a series form without any linearization, discretization, or perturbation. Using convergence analysis, we observed that the proposed method is converging to the exact solution of the system. Numerical analysis for these solutions shows the confine carrier wave and ensures localized wave packets. A degree enhancement reduction in the dispersion nonlinearly coefficients alters both the spatial width and amplitude of the waves. At vanishingly small dispersion, internal oscillations in the pulse-shaped solitary waves reduced the wave breaking effect. We have noticed that the numerical solutions converge to the exact solutions, confirming the importance of the proposed technique.