Fractional Complex Transforms, Reduced Equations And Exact Solutions Of The Fractional Kraenkel–Manna–Merle System

Exact solutions of the fractional Kraenkel–Manna–Merle system in saturated ferromagnetic materials have been studied. Using the fractional complex transforms, the fractional Kraenkel–Manna–Merle system is reduced to ordinary differential equations, (1 + 1)-dimensional partial differential equations and (2 + 1)-dimensional partial differential equations. Based on the obtained ordinary differential equations and taking advantage of the available solutions of Jacobi elliptic equation and Riccati equation, soliton solutions, combined soliton solutions, combined Jacobi elliptic function solutions, triangular periodic solutions and rational function solutions, for the KMM system are obtained. For the obtained (1 + 1)-dimensional partial differential equations, we get the classification of Lie symmetries. Starting from a Lie symmetry, we get a symmetry reduction equation. Solving the symmetry reduction equation by the power series method, power series solutions for the KMM system are obtained. For the obtained (2 + 1)-dimensional partial differential equations, we derive their bilinear form and two-soliton solution. The bilinear form can also be used to study the lump solutions, rogue wave solutions and breathing wave solutions.