Oceanic water waves via double and single convex-concave solitons in the generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt equations with a novel truncated MM-derivative

Abstract A classical system of generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt equations (gWBKBKEs), describing long shallow-water waves in a dispersive medium, has been extensively studied. This paper introduces a novel fractional derivative, the truncated modified Mittag-Leffler function derivative (TMMD), and investigates its impact on behaviors of solutions of gWBKBKEs. Using the unified method (UM), exact solutions for single-wave systems are derived, as the equations are integrable in such cases. However, for two-wave systems, where integrability is lost, only approximate solutions are achievable. The UM ensures that the maximum error is controlled through optimal parameter selection in the remainder terms The horizontal velocity and wave height deviations from the equilibrium are illustrated. Novel waves structures are revealed including convex-cavity, concave-cavity-solitary waves and curly wave-train. Also, a formation of waves in interactions of the basin and solitons is found. These results are explored via the derived solutions. Stability analysis asserts that the initial state becomes unstable when the initial horizontal velocity exceeds a critical threshold linked to the deviation height. Furthermore, the effect of the fractional parameter in the TMMD is examined, showing minimal impact on the intensity of horizontal velocity and wave height deviations.