Ishimori equation

The Ishimori equation has the form {{NumBlk|| \frac{\partial \mathbf{S}}{\partial t} = \mathbf{S}\wedge \left(\frac{\partial^2 \mathbf{S}}{\partial x^2} + \frac{\partial^2 \mathbf{S}}{\partial y^2}\right)+ \frac{\partial u}{\partial x}\frac{\partial \mathbf{S}}{\partial y} + \frac{\partial u}{\partial y}\frac{\partial \mathbf{S}}{\partial x},|}} {{NumBlk|| \frac{\partial^2 u}{\partial x^2}-\alpha^2 \frac{\partial^2 u}{\partial y^2}=-2\alpha^2 \mathbf{S} \cdot \left(\frac{\partial \mathbf{S}}{\partial x}\wedge \frac{\partial \mathbf{S}}{\partial y}\right).|}} ==Lax representation==

The Lax representation of the equation is given by Here {{NumBlk||\Sigma=\sum_{j=1}^3S_j\sigma_j,|}} the \sigma_i are the Pauli matrices and I is the identity matrix. ==Reductions==

The Ishimori equation admits an important reduction: in 1+1 dimensions it reduces to the continuous classical Heisenberg ferromagnet equation (CCHFE). The CCHFE is integrable. ==Equivalent counterpart==

The equivalent counterpart of the Ishimori equation is the Davey-Stewartson equation. ==See also==