# The Landau-Lifshitz equation, the NLS, and the magnetic rogue wave as a by-product of two colliding regular "positons"

## TL;DR

This paper introduces a novel method to generate exact solutions of the Landau-Lifshitz-Gilbert equation by leveraging the nonlinear Schrödinger equation and binary Darboux transformations, enabling the construction of complex rogue wave solutions.

## Contribution

The authors develop a new approach to produce multiple-rogue wave solutions of NLS directly from NLS solutions without relying on KP-I, expanding the set of known solutions including the novel 'impacton' rogue wave.

## Key findings

- Constructed all Dubard-Matveev P-breathers using the new method.
- Discovered a new rogue wave solution called 'impacton' resulting from colliding positon-like waves.
- Demonstrated that the approach can generate previously unknown solutions.

## 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 Schr\"odinger 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".