New interaction and combined multi-wave solutions for the Heisenberg ferromagnetic spin chain equation
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New interaction and combined multi-wave solutions for the Heisenberg ferromagnetic spin chain equation Abdullahi Yusuf1,2 , Fairouz Tchier3 , Mustafa Inc4,5,a 1 2 3 4
Department of Computer Engineering, Biruni University, Istanbul, Turkey Department of Mathematics Jigawa, Faculty of Science, Federal University, Dutse, Dutse, Nigeria Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia Department of Mathematics, Faculty of Science, Firat University, Elazig, Turkey 5 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan Received: 31 December 2019 / Accepted: 28 April 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In the current study, rational, interaction and combined multi-wave solutions are obtained for Heisenberg ferromagnetic spin chain equation by using the logarithmic transformation and symbolic computation with ansatz functions. By the choice of suitable parameter values, the dynamics of these obtained solutions are analyzed and represented in figures. The dynamics and characteristics of lump–kink, soliton and multi-wave interactions observed through graphical analysis.
1 Introduction Rogue waves (RW) are expansive and instinctive ocean waves that have drawn growing focus on both theoretical and experimental observations [1]. The RW for nonlinear Schrodinger equation in simplest form has been proposed in [2]. It can be seen that there are huge wave phenomena in different fields like plasmas, nonlinear optics, Bose–Einstein condensates, biophysics and even finance [3–5]. In terms of a new combination of variable functions using the Hirota bilinear model, some researchers are working out some new solutions from the lump solution family and some groups of interaction solutions. We are reviewing some literature on the phenomena of lump solution and interaction. To this aim, there have been series of presentation of lump solutions from different perspectives, for instant, Zakharov [6], pump wave solution [7], lump solution through Hirota bilinear method [8–11]. Through important properties of lump solutions, it can be understood that amplitudes, shapes, speeds of solitons will be preserved after collision with another soliton and this is the elastic property of collision. Moreover, interaction between rouge wave and kink solitary wave solution has been established in [12–17]. Several other types of solution can also be found in [18–28]. In this study, we want employ the Hirota bilinear scheme to construct some novel rogue and interaction solution for the Heisenberg ferromagnetic spin chain equation (HFSCE) [29] given by iχt + λ1 χx x + λ2 χ yy + λ3 χx y − λ4 |χ|2 χ = 0,
(1)
a e-mail: [email protected] (corresponding author)
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Eur. Phys. J. Plus
(2020) 135:416
where λ4 = 2Bτ 4 , λ1 = (σ + σ2 ) τ 4 , λ2 = (σ1 + σ2 ) τ 4 , λ3 = 2σ2 τ 4 , denotes the lattice parameter, the coefficients of bilinear exchange interact
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