Ishimori system

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

$\left\{{\begin{aligned}iu_{t}+u_{xx}-au_{yy}=2u(u_{x}^{2}-u_{y}^{2})/(1+|u|^{2})-ib(\phi _{x}u_{y}-\phi _{y}u_{x})\\\phi _{xx}+a'\phi _{yy}=8Im(u_{x}u_{y})/(1+|u|^{2})^{2}\end{aligned}}\right.$

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 $H^{4}$ intersect $L^{2}((x^{2}+y^{2})^{4}dxdy)$ KnPoVe2000.