Effective generation of closed-form soliton solutions of the continuous classical Heisenberg ferromagnet equation

Abstract

The non-topological, stationary and propagating, soliton solutions of the classical continuous Heisenberg ferromagnet equation are investigated. A general, rigorous formulation of the Inverse Scattering Transform for this equation is presented, under less restrictive conditions than the Schwartz class hypotheses and naturally incorporating the non-topological character of the solutions. Such formulation is based on a new triangular representation for the Jost solutions, which in turn allows an immediate computation of the asymptotic behaviour of the scattering data for large values of the spectral parameter, consistently improving on the existing theory. A new, general, explicit multi-soliton solution formula, amenable to computer algebra, is obtained by means of the matrix triplet method, producing all the soliton solutions (including breather-like and multipoles), and allowing their classification and description.

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

“…Here A is an ×n matrix whose n eigenvalues a j n j=1 are obtained from the poles ia j n j=1 of the transmission coefficient T (λ) (namely the discrete eigenvalues) by multiplication by a factor −i (a proof of this fact can be found in [25]); B is ā n × 1 matrix; and C is a 1 ×n matrix. Furthermore, we assume that the triplet (A, B, C) is a minimal triplet in the sense that the matrix order of A is minimal among all triplets representing the same Marchenko kernel by means of (13) [37,38].…”

“…We note that formulae (22) give the soliton solutions of (1a) with the so-called easy-axis conditions (i.e., m(x) → e 3 as x → ±∞, see [25]). Thus, equations (21) allow one to generate the soliton solutions of (1a) with boundary conditions (1b) when a soliton solutions of (1a) with easy-axis conditions is known.…”

“…In this section we construct an explicit soliton solution formula for equation 1aunder the asymptotic condition (1b). To this aim, we apply the IST method (see, for instance, [28][29][30] for more details on this method) combined with the matrix triplet technique, successfully used in [31][32][33][34][35] and more recently in [25] in the context of the classical Heisenberg ferromagnet equation.…”

“…In this spirit, the present work aims at extending the analysis carried out in [25] for the classical, continuous Heisenberg ferromagnet equation with perpendicular ("easy-axis") asymptotic conditions, m(x) → e 3 as x → ±∞, by constructing a new, general formula which generates all reflectionless solutions of (1a) under condition (1b), allowing their classification.…”

Fifth-order generalized Heisenberg supermagnetic models

“…A variety of the work has focused on the investigation of the generalized HF models, such as higher-order deformations of HF models [9,10], the multi-component generalized HF models [11], the multidimensional extended HF models [12,13]. Later on the N-soliton solutions of the generalized HF models have also been analyzed [7,14].…”