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.
Highlights
A number of theoretical and mostly experimental advancements have ignited renewed interest toward the study of propagating magnetic-droplet soliton aDipartimento di Matematica e Informatica, Universita degli Studi di Cagliari, Viale Merello 92, 09121 Cagliari, Italy bDepartment of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, United KingdomThe experimental observation of solitons and solitary waves in ferromagnetic systems has proved challenging, mainly due to the dimensions of the length-scale at which these phenomena are expected to occur [35, 38]
The second objective of this paper is to find a general, explicit multi-soliton solution formula for (1.2)
In Appendix A we determine the Marchenko equations by using the triangular representation introduced in Section 2, and in Appendix B we give further details of the derivation of the solution formula and we provide alternative formulations of it
Summary
A number of theoretical and mostly experimental advancements have ignited renewed interest toward the study of propagating (non-topological) magnetic-droplet soliton aDipartimento di Matematica e Informatica, Universita degli Studi di Cagliari, Viale Merello 92, 09121 Cagliari, Italy bDepartment of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, United Kingdom. On a more general note, it is worth clarifying that, even if the Riemann-Hilbert problem for Zakharov-Shabat systems as well as the reflectionless solutions of the nonlinear Schrodinger equation have been extensively investigated for a very long time (e.g., see [2, 13, 25, 58, 1, 20]), the gauge equivalence [61] does not automatically entail that from there one can and immediately recover a general, explicit, multi-soliton solution formulae for (1.2) (see [8]) To obtain this result we will develop the matrix triplet method, already employed to solve exactly, in the reflectionless case, several other integrable equations (e.g., see [6, 4, 19, 5, 23, 21]). In Appendix A we determine the Marchenko equations by using the triangular representation introduced in Section 2, and in Appendix B we give further details of the derivation of the solution formula and we provide alternative (and more explicit) formulations of it
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