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Symmetry 2018, 10(4), 82; https://doi.org/10.3390/sym10040082

The Landau-Lifshitz Equation, the NLS, and the Magnetic Rogue Wave as a By-Product of Two Colliding Regular “Positons”

1
Department of Physics, Mathematics and Informational Technology, Immanuel Kant Baltic Federal University, Al. Nevsky St. 14, 236041 Kaliningrad, Russia
2
Functionalized Magnetic Materials for Biomedicine and Nanotechnology Center, Department of Physics, Mathematics and Informational Technology, Immanuel Kant Baltic Federal University, Mathematics and IT, Al. Nevsky St. 14, 236041 Kaliningrad, Russia
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Received: 16 February 2018 / Revised: 18 March 2018 / Accepted: 19 March 2018 / Published: 27 March 2018
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
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Abstract

In this article we present a new method for construction of exact solutions of the Landau-Lifshitz-Gilbert equation (LLG) for ferromagnetic nanowires. The method is based on the established relationship between the LLG and the nonlinear Schrödinger equation (NLS), and is aimed at resolving an old problem: how to produce multiple-rogue wave solutions of NLS using just the Darboux-type transformations. The solutions of this type—known as P-breathers—have been proven to exist by Dubard and Matveev, but their technique heavily relied on using the solutions of yet another nonlinear equation, the Kadomtsev-Petviashvili I equation (KP-I), and its relationship with NLS. We have shown that in fact one doesn’t have to use KP-I but can instead reach the same results just with NLS solutions, but only if they are dressed via the binary Darboux transformation. In particular, our approach allows us to construct all the Dubard-Matveev P-breathers. Furthermore, the new method can lead to some completely new, previously unknown solutions. One particular solution that we have constructed describes two “positon”-like waves, colliding with each other and in the process producing a new, short-lived rogue wave. We called this unusual solution (in which a rogue wave is begotten after the impact of two solitons) the “impacton”. View Full-Text
Keywords: Landau-Lifshitz-Gilbert equation; nonlinear Schrödinger equation; Darboux transformation; P-breathers; positons Landau-Lifshitz-Gilbert equation; nonlinear Schrödinger equation; Darboux transformation; P-breathers; positons
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Yurov, A.V.; Yurov, V.A. The Landau-Lifshitz Equation, the NLS, and the Magnetic Rogue Wave as a By-Product of Two Colliding Regular “Positons”. Symmetry 2018, 10, 82.

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