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