Dynamics of Damped and Undamped Wave Natures of the Fractional Kraenkel-Manna-Merle System in Ferromagnetic Materials

Document Type : Research Paper


1 Department of Mathematics, Pabna University of Science and Technology, Pabna, 6600, Bangladesh

2 Department of Computer Science and Engineering, Pabna University of Science and Technology, Pabna, 6600, Bangladesh

3 Department of Electrical, Electronic and Communication Engineering, Pabna University of Science and Technology, Pabna, 6600, Bangladesh

4 Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Van, Turkey


This research considers the Kraenkel-Manna-Merle system with an M-truncated derivative (K-M-M-S-M-T-D) that defines the magnetic field propagation (M-F-P) in ferromagnetic materials with zero conductivity (F-M-Z-C) and uses the Sardar sub-equation method (S-S-E-M). Our goal is to acquire soliton solutions (SSs) of K-M-M-S-M-T-D via the S-S-E-M. To our knowledge, no one has considered the SSs to the K-M-M-S-MTD with or without a damping effect (DE) via the S-S-E-M. The SSs are achieved as the M-shape, periodic wave shape, W-shape, kink, anti-parabolic, and singular kink solitons in terms of free parameters. We utilize Maple to expose pictures in three-dimensional (3-D), contour and two-dimensional (2-D) for different values of fractional order (FO) of the got SSs, and we discuss the effect of the FO of the K-M-M-S-MTD via the S-S-E-M, which has not been discussed in the previous literature. All wave phenomena are applied to optical fiber communication, signal transmission, porous mediums, magneto-acoustic waves in plasma, electromagnetism, fluid dynamics, chaotic systems, coastal engineering, and so on. The achieved SSs prove that the S-S-E-M is very simple and effective for nonlinear science and engineering for examining nonlinear fractional differential equations (N-L-F-D-Es).


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Iqbal, N., Wu, R., Mohammed, W.W., Pattern formation induced by fractional cross-diffusion in a 3-species food chain model with harvesting, Mathematics and Computers in Simulation, 188, 2021, 102-119.
[2] Islam, S., Alam, M.N., Asad, M.F.A., Tunc, C., An analytical technique for solving new computational solutions of the modified Zakharov-Kuznetsov equation arising in electrical engineering, Journal of Applied and Computational Mechanics, 7(2), 2021, 715-726.
[3] Alam, M.N., Osman, M.S., New structures for closed-form wave solutions for the dynamical equations model related to the ion sound and Langmuir waves, Communications in Theoretical Physics, 73(3), 2021, 035001.
[4] Alam, M.N., Li, X., New soliton solutions to the nonlinear complex fractional Schrödinger equation and the conformable time-fractional Klein–Gordon equation with quadratic and cubic nonlinearity, Physica Scripta, 95, 2020, 045224.
[5] Dietl, T., Sato, K., Fukushima, T., Bonanni, A., Jamet, M., Barski, A., Kuroda, S., Tanaka, M., Hai, P.N., Yoshida, H.K., Spinodal nanodecomposition in semiconductors doped with transition metals, Reviews of Modern Physics, 87, 2015, 1311-1377.
[6] Shen, B.G., Sun, J.R., Hu, F.X., Zhang, H.W., Cheng, Z.H., Recent progress in exploring magnetocaloric materials, Advanced Materials, 21, 2009, 4545–4564.
[7] Tanaka, M., Ohya, S., Hai, P.N., Recent progress in III–V based ferromagnetic semiconductors: Band structure, Fermi level, and tunneling transport, Applied Physics Reviews, 1, 2014, 011102.
[8] Das, N., Ray, S.S., Exact traveling wave solutions and soliton solutions of conformable M-fractional modified nonlinear Schrödinger model, Optik, 287, 2023, 171060.
[9] Alsharidi, A.K., Bekir, A., Discovery of new exact wave solutions to the M-fractional complex three coupled Maccari’s system by Sardar sub-equation scheme, Symmetry, 15, 2023, 1567.
[10] Faisal, K., Abbagari, S., Pashrashid, A., Houwe, A., Yao, S.W., Ahmad, H., Pure-cubic optical solitons to the Schrödinger equation with three forms of nonlinearities by Sardar subequation method, Results in Physics, 48, 2023; 106412.
[11] Asghari, Y., Eslami, M., Rezazadeh, H., Exact solutions to the conformable time-fractional discretized mKdv lattice system using the fractional transformation method, Optical and Quantum Electron, 55, 2023, 318.
[12] Hong, B., Exact solutions for the conformable fractional coupled nonlinear Schrödinger equations with variable coefficients, Journal of Low Frequency Noise, Vibration & Active Control, 42(2), 2023, 628-641.
[13] Yin, Q.,  Gao, B., Shi, Z., Distinct exact solutions for the conformable fractional derivative Gerdjikov-Ivanov equation via three credible methods, Journal of Taibah University for Science, 17, 2023, 2251219.
[14] Ahmad, H., Alam, M.N., Rahman, M.A., Alotaibid, M.F., Omri, M., The unified technique for the nonlinear time-fractional model with the beta-derivative, Results in Physics, 29, 2021, 104785.
[15] Ullah, M.S., Roshid, H.O., Ali, M.Z., New wave behaviors of the Fokas-Lenells model using three integration techniques, PloS One, 18(9), 2023, e0291071. 
[16] Ullah, M.S., Mostafa, M., Ali, M.Z., Roshid, H.O., Akter, M., Soliton solutions for the Zoomeron model applying three analytical techniques, PloS One, 18(7), 2023, e0283594.
[17] Alam, M.N., Islam, S.M.R., The agreement between novel exact and numerical solutions of nonlinear models, Partial Differential Equations in Applied Mathematics, 8,  2023, 100584.
[18] Alam, M.N., An analytical technique to obtain traveling wave solutions to nonlinear models of fractional order, Partial Differential Equations in Applied Mathematics, 8, 2023, 100533. 
[19] Alam, M.N., Soliton solutions to the electric signals in telegraph lines on the basis of the tunnel diode, Partial Differential Equations in Applied Mathematics, 7, 2023, 100491.
[20] Alam, M. N., Talib, I., Tunc, C., The new soliton configurations of the 3D fractional model in arising shallow water waves, International Journal of Applied and Computational Mathematics, 9, 2023, 75.
[21] Alam, M.N., Akash, H.S., Saha, U., Hasan, M.S., Parvin, M.W., Tunç, C., Bifurcation Analysis and Solitary Wave Analysis of the Nonlinear Fractional Soliton Neuron Model, Iranian Journal of Science, 2023, DOI: https://doi.org/10.1007/s40995-023-01555-y.
[22] Alam, M.N., Exact solutions to the foam drainage equation by using the new generalized (G’/G)-expansion method, Results in Physics, 5, 2015, 168-177.
[23] Alam, M.N., İlhan, O.A., Uddin, M.S., Rahim, M.A., Regarding on the results for the Fractional Clannish Random Walker’s Parabolic equation and the nonlinear fractional Cahn-Allen equation, Advances in Mathematical Physics, 2022, 2022, 5635514.
[24] Nguepjouo, F.T., Kuetche, V.K., Kofane, T.C., Soliton interactions between multivalued localized waveguide channels within ferrites, Physical Review E, 89, 2014, 063201.
[25] Alshammari, M., Hamza, A.E., Cesarano, C., Aly, E.S., Mohammed, W.W., The Analytical Solutions to the Fractional Kraenkel–Manna–Merle System in Ferromagnetic Materials, Fractal and Fractional, 7, 2023, 523.
[26] Li, B.Q., Ma, Y.L., Rich soliton structures for the Kraenkel-Manna-Merle (KMM) System in Ferromagnetic Materials, Journal of Superconductivity and Novel Magnetism, 31, 2018, 1773–1778.
[27] Li, B.Q., Ma, Y.L., Oscillation rogue waves for the Kraenkel–Manna–Merle system in ferrites, Journal of Magnetism and Magnetic Materials, 537, 2021, 168182.
[28] Tripathy, A., Sahoo, S., Rezazadeh, H., Izgi, Z.P., Osman, M.S., Dynamics of damped and undamped wave natures in ferromagnetic materials, Optik, 281, 2023, 170817.
[29] Raza, N., Hassan, Z., Butt, A.R., Rahman, R.U., Aty, A.H.A., Mahmoud, M., New and more dual-mode solitary wave solutions for the Kraenkel–Manna–Merle system incorporating fractal effects, Mathematical Methods in the Applied Sciences, 45, 2022, 2964-2983.
[30] Zhang, L., Shen, B., Jiao, H., Wang, G., Wang, Z., Exact solutions for the KMM system in (2+1)‐dimensions and its fractional form with beta‐derivative, Fractal and Fractional, 6, 2022, 520.
[31] Jin, X.W., Lin, J., The contributions of Gilbert-damping and inhomogeneous exchange effects on the electromagnetic short waves propagation in saturated ferrite films, Journal of Magnetism and Magnetic Materials, 514, 2020, 167192.
[32] Ma, Y.L., Li, B.Q., Kraenkel-Manna-Merle saturated ferromagnetic system: Darboux transformation and loop-like soliton excitations, Chaos, Solitons and Fractals, 159, 2022, 112179.
[33] Tchokouansi, H.T., Felenou, E.T., Kuetche, V.K., Tchidjo, R.T., Dynamics of damped single valued magnetic wave in inhomogeneous circularly polarized ferrites, Chinese Journal of Physics, 78, 2022, 511-520.
[34] Shen, S.J., Li, H.J., Lin, J., Propagations of multiple solitons in a deformed ferrite, Results in Physics, 51, 2023, 106645.
[35] Jin, X.W., Lin, J., Rogue wave, interaction solutions to the KMM system, Journal of Magnetism and Magnetic Materials, 502, 2020, 166590.
[36] Li, B.Q., Ma, Y.L., Sathishkumar, P., The oscillating solitons for a coupled nonlinear system in nanoscale saturated ferromagnetic materials, Journal of Magnetism and Magnetic Materials, 474, 2019, 661-665.
[37] Mohammed, W.W., El-Morshedy, M., Cesarano, C., Al-Askar, F.M., Soliton Solutions of Fractional Stochastic Kraenkel–Manna– Merle Equations in Ferromagnetic Materials, Fractal and Fractional, 7, 2023, 328.
[38] Li, B.Q., Ma, Y.L., Loop-like periodic waves and solitons to the Kraenkel–Manna–Merle system in ferrites, Journal of Electromagnetic Waves and Applications, 32, 2018, 1275–1286.
[39] Sousa, J.V., Oliveira, E.C.D., A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, International Journal of Analysis and Applications, 16, 2018, 8396.