Electron. J. Differential Equations, Vol. 2018 (2018), No. 106, pp. 1-34.

Heisenberg ferromagnetism as an evolution of a spherical indicatrix: localized solutions and elliptic dispersionless reduction Francesco Demontis, Giovanni Ortenzi, Matteo Sommacal

Abstract:
A geometrical formulation of Heisenberg ferromagnetism as an evolution of a curve on the unit sphere in terms of intrinsic variables is provided and investigated. Given a vortex filament moving in an incompressible Euler fluid with constant density (under the local induction approximation hypotheses), the solutions of the classical Heisenberg ferromagnet equation are represented by the corresponding spherical (or tangent) indicatrix. The equations for the time evolution of the indicatrix on the unit sphere are given explicitly in terms of two intrinsic variables, the geodesic curvature and the arc-length of the curve. Notably, by considering the evolution with respect to slow variables and neglecting the dispersive terms, a novel elliptic dispersionless reduction of the Heisenberg ferromagnet model is obtained. The length of the spherical indicatrix is proved not to be conserved. Finally, a totally explicit algorithm is provided, allowing to construct a solution of the Heisenberg ferromagnet equation from a solution of Nonlinear Schrodinger equation, and, remarkably, viceversa, allowing to construct a solution of Nonlinear Schrodinger equation from a solution of the Heisenberg ferromagnet equation. As expected from the Zakharov-Takhtajan gauge equivalence, in the reflectionless case such a two-way map between solutions is shown to preserve the Inverse Scattering Transform spectra, and thus the localization.

Submitted January 17, 2018. Published May 7, 2018.
Math Subject Classifications: 35C08, 35J62, 35Q35, 35Q51, 53C44.
Key Words: Classical Heisenberg ferromagnet equation; curve motion; nonlinear Schroedinger equation; vortex filament binormal motion; dispersionless reduction; localized solution.

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 Francesco Demontis Dipartimento di Matematica e Informatica Universitá di Cagliari 09124 Cagliari, Italy email: fdemontis@unica.it Giovanni Ortenzi Dipartimento di Matematica Pura e Applicazioni Universitá di Milano Bicocca 20125 Milano, Italy email: giovanni.ortenzi@unimib.it Matteo Sommacal Department of Mathematics Physics and Electrical Engineering Northumbria University Newcastle upon Tyne, NE1 8ST, UK email: matteo.sommacal@northumbria.ac.uk