FRACTIONAL COMPLEX TRANSFORMS, REDUCED EQUATIONS AND EXACT SOLUTIONS OF THE FRACTIONAL KRAENKEL–MANNA–MERLE SYSTEM

Lihua Zhang, Zhenli Wang and Bo Shen
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Lihua Zhang: School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang 050061, P. R. China
Zhenli Wang: ��School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, P. R. China
Bo Shen: School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang 050061, P. R. China

FRACTALS (fractals), 2022, vol. 30, issue 09, 1-15

Abstract: 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.

Keywords: Fractional KMM System; Lie Symmetry; Exact Solutions; Power Series Solutions; Soliton Solutions; Bilinear Form; Jacobi Elliptic Equation; Riccati Equation (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22501791

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