Exact Solutions for the KMM System in (2+1)-Dimensions and Its Fractional Form with Beta-Derivative

• Published in: Fractal and fractional Vol. 6; no. 9; p. 520
• Main Authors: Zhang, Lihua, Shen, Bo, Jiao, Hongbing, Wang, Gangwei, Wang, Zhenli
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-09-2022
• Subjects: Algebra; conservation laws; Derivatives; Differential equations; Elliptic functions; Exact solutions; Ferromagnetic materials; Fractional calculus; KMM system; Lie symmetries; optimal system; Ordinary differential equations; Partial differential equations; Power series; power series solutions; Rational functions; Reduction; Solitary waves; Symmetry; Transformations (mathematics); Traveling waves; Wave propagation
• DOI: 10.3390/fractalfract6090520
• EISSN: 2504-3110

Fractional calculus is useful in studying physical phenomena with memory effects. In this paper, the fractional KMM (FKMM) system with beta-derivative in (2+1)-dimensions was studied for the first time. It can model short-wave propagation in saturated ferromagnetic materials, which has many applications in the high-tech world, especially in microwave devices. Using the properties of beta-derivatives and a proper transformation, the FKMM system was initially changed into the KMM system, which is a (2+1)-dimensional generalization of the sine-Gordon equation. Lie symmetry analysis and the optimal system for the KMM system were investigated. Using the optimal system, we obtained eight (1+1)-dimensional reduction equations. Based on the reduction equations, new soliton solutions, oblique analytical solutions, rational function solutions and power series solutions for the KMM system and FKMM system were derived. Using the properties of beta-derivatives and another transformation, the FKMM system was changed into a system of ordinary differential equations. Based on the obtained system of ordinary differential equations, Jacobi elliptic function solutions and solitary wave solutions for the FKMM system were derived. For the KMM system, the results about Lie symmetries, optimal system, reduction equations, and oblique traveling wave solutions are new, since Lie symmetry analysis method has not been applied to such a system before. For the FKMM system, all of the exact solutions are new. The main novelty of the paper lies in the fact that beta-derivatives have been used to change fractional differential equations into classical differential equations. The technique can also be extended to other fractional differential equations.