Abundant explicit solutions for the M-fractional coupled nonlinear Schrödinger–KdV equations

In this study, the famous fractional generalized coupled cubic nonlinear Schrödinger–KdV equations arising in many domains of physics and engineering such as depicting the propagation of long waves in dispersive media and the dynamics of short dispersive waves for narrow-bandwidth packet have been investigated. We propose two significant methods named the modified (Gʹ/G, 1/G)-expansion method and the Gʹ/(bGʹ + G + a)-expansion method. After utilizing these two efficient techniques, many types of explicit soliton pulse solutions including the well-known bell-shape bright soliton pulse, the kink and anti-kink dark solitons pulse, the mixed bright-dark soliton pulse, the W-shaped soliton pulse, the periodic waves pulse, and the blow-up soliton pattern solutions are obtained. If we select different values of the frequency, coefficients and orders, the dynamic properties and physical structures of these solutions are discussed, these important results can help us to further understand the inner characteristic of the model.