Ishimori system
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The Ishimori system [Im1984] has a complex field u and a real field phi in two dimensions, and has the form
iu_t + u_xx - a u_yy = 2 u (u_x^2 - u_y^2) / (1 + |u|^2) - i b(phi_x u_y - phi_y u_x) phi_xx + a' phi_yy = 8 Im(u_x u_y) / (1 + |u|^2)^2
The case (a,a') = (+1,-1) is studied in [Sy1992]. The case (a,a') = (-1,1) is studied in [HySau1995], [Ha-p], [KnPoVe2000]; in this case one has LWP for small data in the space H^4 intersect L^2( (x^2 + y^2)^4 dx dy) [KnPoVe2000].
