Comprehensive analytical treatment of bifurcation, stability, and multiform solitons for nonlinear conformable evolution equation in shallow-water waves

Abstract

We apply the modified extended (ME) mapping algorithm to the combined Shallow Water Wave-Kadomtsev-Petviashvili equations in a (2+1)-dimensional framework involving the conformable derivative to construct new analytic solutions. Compared with classical techniques such as the Hirota bilinear method and Jacobi elliptic function expansions, the ME mapping algorithm offers greater flexibility and produces a broader spectrum of wave structures for the fractional model. Accordingly, several solution families are obtained, including bright, dark, and singular solitons, periodic and singular periodic waves, Weierstrass doubly periodic solutions, hyperbolic waves, Jacobi elliptic solutions, and exponential-type structures. Notably, mixed elliptic-exponential forms, certain singular periodic waves, and specific dark soliton configurations are reported for the first time for this system. A detailed bifurcation and stability analysis reveals rich dynamical behavior, including collision-free and recurrent soliton interactions under appropriate parameter regimes. Two- and three-dimensional graphical representations illustrate solution morphology and evolution, with potential physical applications.