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On the initial value problem for the Ishimori system. (English) Zbl 1021.35105
The authors study the Ishimori system $$\align & \partial_tS = S\wedge (\partial^2_xS\pm\partial^2_y S)+b(\partial_x\phi\partial_y S+\partial_y\phi\partial_x S),\quad t\in\bbfR,\ x,y\in\bbfR,\\ & \partial^2_x\phi \mp \partial^2_y\phi = \mp 2S\cdot (\partial_x S\wedge \partial_y S),\endalign$$ where $S(\cdot,t) : \bbfR^2 \to \bbfR^3$ with $\|S\|= 1$, $S\to (0,0,1)$ as $\|(x,y)\|\to\infty$, and $\wedge$ denotes the wedge product in $\bbfR^3$. 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.

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)
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