Reliable methods to study some nonlinear conformable systems in shallow water

Abstract

AbstractIn this paper, consider the eminent coupled Boussinesq–Burger (BB) equations and the coupled Whitham–Broer–Kaup (WBK) equations with time fractional derivative arising in the investigation of shallow water waves. The derivative is described in the sense of conformable derivative. We introduce the fundamental ${(G'} / {G} )$ ( G ′ / G ) -expansion method and its extension, namely the two-variable ${(G'} / {G}, {1} / {G} )$ ( G ′ / G , 1 / G ) -expansion method, to establish general solutions, some typical wave solutions existing in the literature, and some new and compatible soliton solutions comprised with certain parameters. For the definite values of these parameters, we derive and show in figures the well-known kink, singular kink, bell-shape soliton, periodic soliton, cuspon, and so on. The obtained solutions affirm that the introduced methods are reliable and efficient techniques to examine a wide variety of nonlinear fractional systems in the sense conformable derivative.