The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. II. IST and closed-form soliton solutions

Abstract

A new, general, closed-form soliton solution formula for the classical Heisenberg ferromagnet equation with in-plane asymptotic conditions is obtained by means of the Inverse Scattering Transform (IST) technique and the matrix triplet method. This formula encompasses the soliton solutions already known in the literature as well as a new class of soliton solutions (the so-called multipole solutions), allowing their classification and description. Examples from all classes are provided and discussed.

The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. I. Direct and inverse scattering theory

“…Following, for instance, [11,14], we formulate and solve this problem by using the Marchenko method. First of all we prove the following Theorem 3 The auxiliary function K up (x, y) which appears in (19) has to satisfy the following integral Marchenko equations:…”

“…From (19), it is immediate to see that e −iλx ψ(x, λ) is continuous in λ ∈ C + , is analytic in λ → C + , and tends to the first column of H up (x) as λ → ∞ from within C + . Analogously, e iλx ψ(x, λ) is continuous in λ ∈ C − , is analytic in λ ∈ C − , and tends to the second column of H up (x) as λ → ∞ from within C − .…”

“…Consequently, this work will pave the way to the investigation of the reflectionless solutions also for this latter equation by means of the IST machinery (which is based on the direct and inverse scattering theory) and the "triplet method" (see, for example, [13][14][15][16][17][18] for a detailed description of the triplet method). We postpone the actual development of the IST and the generation of closed-form soliton solutions for (2a) with (2b) to the second part of this work [19].…”