An efficient and highly accurate spectral method for modeling the propagation of solitary magnetic spin waves in thin fi
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An efficient and highly accurate spectral method for modeling the propagation of solitary magnetic spin waves in thin films Marios A. Christou1 · Nectarios C. Papanicolaou1 · Christodoulos Sophocleous2 Received: 6 February 2020 / Revised: 6 February 2020 / Accepted: 15 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In previous works, it was shown that the propagation of magnetic spin waves in thin films can be approximated by a nonlinear Schrödinger-type equation. The formulation begins with the magnetostatic equations (Gauss and Ampere’s laws of magnetism) and the Landau–Lifshitz equation. The solution of this system is a potential function whose dimensionless amplitude is the solution of a nonlinear Schrödinger. In the current work, we are demonstrating an efficient infinite series solution using the Christov functions. This is the first time the functions are used in problems involving complex arithmetics. The solutions of the time-independent and time-dependent problems are given in complex series form. Keywords Magnetic thin solitons · Spectral method · Galerkin Mathematics Subject Classification 35J10 · 65M70 · 81V60
1 Introduction Solitary waves were discovered in the mid-nineteenth century in the form of localized persistent shallow water waves, in what now has become a famous account by Russell (1845). Following this initial discovery, Boussinesq (1871a, b) proved that permanent waves exist in nonlinear systems as a result of the balance between nonlinearity and dispersion. A few decades later, Zabusky and Kruskal (1965) used numerical methods to discover wave solu-
Communicated by Pierangelo Marcati.
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Marios A. Christou [email protected] Nectarios C. Papanicolaou [email protected] Christodoulos Sophocleous [email protected]
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Department of Mathematics, University of Nicosia, Nicosia, Cyprus
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Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus 0123456789().: V,-vol
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tions of the well-known Korteweg–de Vries equation which exhibited particle-like behavior. They named these solutions solitons, a term alluding to a subatomic particle and encompassing the notion of a solitary wave. Ever since, solitons have been studied extensively both theoretically and experimentally and many important results have been published. Their applications have proven to be multifaceted and wide-ranging, with various areas in Physics, Engineering and even Biology using solitons as models for many different processes (Lee 2006; Scott 1992; Georgiev and Glazebrook 2019; Chabchoub et al. 2013; Remoissenet 1999; Trines et al. 2007; Deng et al. 2019). One of the most important equations that admit solitary wave solutions is the classic nonlinear Schrödinger equation (Zakharov 1968) and its many variants and generalizations. This class of equations and systems has attracted the interest of scientists from many different disciplines. It was shown that in its various versions it successfully mo
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