## Smooth solution of the generalized system of ferro-magnetic chain.(English)Zbl 1039.82037

Summary: The authors consider the initial value problem with periodic boundary condition for a class of general systems of the ferromagnetic chain $z_t=-\alpha z\times (z\times z_{xx}) +z\times z_{xx}+ z\times f(x),\quad (\alpha \geq 0).$ The existence of unique smooth solutions is proved by using the technique of spatial difference and a priori estimates of higher-order derivatives in Sobolev spaces.

### MSC:

 82D40 Statistical mechanics of magnetic materials 35Q60 PDEs in connection with optics and electromagnetic theory 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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