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”.
Highlights
The Darboux transformation (DT) [1,2] for one-dimensional Schrödinger equation serves as an important case of so-called isospectral symmetries, which are directly related to the factorisation method [3] and are used to construct the “integrable” potentials from pre-existing ones
Each one of them would have to be constructed with a spectral restriction (43) in mind; and it can be shown that, regardless of the sign in (43), the resulting Peregrine solutions will always be linearly dependent, producing nothing but zero after the second iteration of DT. One way around this obstacle has been proposed by Dubard and Matveev in [29] where they have used the relationship existing between the focusing nonlinear Schrödinger equation (NLS) and the Kadomtsev-Petviashvili I (KP-I)
Such solutions were obtained in a set of articles via some generalization of DT, namely via the Darboux transformations complemented by the differentiation with respect to a spectral parameter
Summary
The Darboux transformation (DT) [1,2] for one-dimensional Schrödinger equation serves as an important case of so-called isospectral symmetries, which are directly related to the factorisation method [3] and are used to construct the “integrable” potentials from pre-existing ones. Until recently all the attempts to construct such solutions have been nothing but exercises in futility, as it soon became clear that the standard techniques—such as the Darboux transformation (see Sections 2 and 4 for more information)—fail to produce any Peregrine soliton-like solutions past the already known ones (the exact reasons for this will be discussed in Sections 3 and 4) This vicious cul-de-sac has only been broken in [28], published in 2010, which proposed a way to construct the multi-rogue waves solution by essentially going around the obstacles and working not with NLS, but with another equation, Kadomtsev-Petviashvili I (KP-I), and afterwards using the relationship between NLS and KP-I to get the required NLS solutions. One interesting aspect of the impacton model lies in slowness of movement of its parental solitons which might make this solution a very good candidate for observation on particular ferromagnetic nanowires
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