The Landau-Lifshitz equation, the NLS, and the magnetic rogue wave as a by-product of two colliding regular "positons"
arXiv:1701.04903 · doi:10.3390/sym10040082
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, 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 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 (rogue wave begotten after the impact of two solitons) the "impacton".
25 pages, 9 figures. Added Section 7 ("7. One last remark: But what of generalization?.."), corrected a number of typos, added 2 more references
References in corpus (1)
Cited by in corpus (4)
- The effect of a small loss or gain in the periodic NLS anomalous wave dynamics. I
- The Cauchy problem for the generalized hyperbolic Novikov-Veselov equation via the Moutard symmetries
- On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation
- The linear and nonlinear instability of the Akhmediev breather