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On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schr\"odinger equations
Using a moving space curve formalism, geometrical as well as gauge
equivalence between a (2+1) dimensional spin equation (M-I equation) and the
(2+1) dimensional nonlinear Schr\"odinger equation (NLSE) originally discovered
by Calogero, discussed then by Zakharov and recently rederived by Strachan,
have been estabilished. A compatible set of three linear equations are obtained
and integrals of motion are discussed. Through stereographic projection, the
M-I equation has been bilinearized and different types of solutions such as
line and curved solitons, breaking solitons, induced dromions, and domain wall
type solutions are presented. Breaking soliton solutions of (2+1) dimensional
NLSE have also been reported. Generalizations of the above spin equation are
discussed.Comment: 32 pages, no figures, accepted for publication in J. Math. Phy
Integrable (2+1)-Dimensional Spin Models with Self-Consistent Potentials
Integrable spin systems possess interesting geometrical and gauge invariance
properties and have important applications in applied magnetism and
nanophysics. They are also intimately connected to the nonlinear Schr\"odinger
family of equations. In this paper, we identify three different integrable spin
systems in (2 + 1) dimensions by introducing the interaction of the spin field
with more than one scalar potential, or vector potential, or both. We also
obtain the associated Lax pairs. We discuss various interesting reductions in
(2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear
Schr\"odinger family of equations, including the (2 + 1)-dimensional version of
nonlinear Schr\"odinger--Hirota--Maxwell--Bloch equations, along with their Lax
pairs.Comment: 21 page
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