# Ishimori equation

The Ishimori equation (IE) is a partial differential equation proposed by the Japanese mathematician Ishimori (1984). Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable Sattinger, Tracy & Venakides (1991, p. 78).

## Equation

The Ishimori equation has the form

${\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \left({\frac {\partial ^{2}\mathbf {S} }{\partial x^{2}}}+{\frac {\partial ^{2}\mathbf {S} }{\partial y^{2}}}\right)+{\frac {\partial u}{\partial x}}{\frac {\partial \mathbf {S} }{\partial y}}+{\frac {\partial u}{\partial y}}{\frac {\partial \mathbf {S} }{\partial x}},\qquad (1a)$
${\frac {\partial ^{2}u}{\partial x^{2}}}-\alpha ^{2}{\frac {\partial ^{2}u}{\partial y^{2}}}=-2\alpha ^{2}\mathbf {S} \cdot \left({\frac {\partial \mathbf {S} }{\partial x}}\wedge {\frac {\partial \mathbf {S} }{\partial y}}\right).\qquad (1b)$

## Lax representation

$L_{t}=AL-LA\qquad (2)$

of the equation is given by

$L=\Sigma \partial _{x}+\alpha I\partial _{y},\qquad (3a)$
$A=-2i\Sigma \partial _{x}^{2}+(-i\Sigma _{x}-i\alpha \Sigma _{y}\Sigma +u_{y}I-\alpha ^{3}u_{x}\Sigma )\partial _{x}.\qquad (3b)$

Here

$\Sigma =\sum _{j=1}^{3}S_{j}\sigma _{j},\qquad (4)$

the $\sigma _{i}$  are the Pauli matrices and $I$  is the identity matrix.

## Reductions

IE 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 IE is the Davey-Stewartson equation.