Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels

In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $0<\alpha \le 1$, which is approaching to the exact solutions when $\alpha =1$. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.