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Zbl 1021.35105
Kenig, C.E.; Ponce, Gustavo; Vega, Luis
On the initial value problem for the Ishimori system. (English)
[J] Ann. Henri Poincaré 1, No.2, 341-384 (2000). ISSN 1424-0637; ISSN 1424-0661/e

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.
[A.D.Osborne (Keele)]

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
60K35 Interacting random processes
82D40 Magnetic materials
82C21 Dynamic continuum models
82C22 Interacting particle systems
82D20 Solids

Keywords: Ishimori system; Heisenberg equation; ferromagnetism; local well-posedness

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