Ishimori system

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The Ishimori system Im1984 has a complex field u and a real field φ in two dimensions, and has the form

\left\{\begin{align} 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 \end{align}\right.

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 H4 intersect L2((x2 + y2)4dxdy) KnPoVe2000.

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