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On the initial value problem for the Ishimori system. (English) Zbl 1021.35105

The authors study the Ishimori system

t S=S( x 2 S± y 2 S)+b( x φ y S+ y φ x S),t,x,y, x 2 φ y 2 φ=2S·( x S y S),

where S(·,t): 2 3 with S=1, S(0,0,1) as (x,y), and denotes the wedge product in 3 . This model was proposed by Y. Ishimori as a two-dimensional generalization of the Heisenberg equation in ferromagnetism, which corresponds to the case b=0 and signs (-,+,+). Their main result shows that, subject to certain conditions, there exists a unique solution to an associated initial value problem, so showing the local well-posedness of this associated problem, with data of arbitrary size in a weighted Sobolev space.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
60K35Interacting random processes; statistical mechanics type models; percolation theory
82D40Magnetic materials (statistical mechanics)
82C21Dynamic continuum models (systems of particles, etc.)
82C22Interacting particle systems
82D20Solids (statistical mechanics)