# Ishimori equation

The Ishimori equation (IE) is a partial differential equation proposed by the Japanese mathematician Yuji 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

$L=\Sigma \partial_x+\alpha I\partial_y,\qquad (3a)$
$A= -2i\Sigma\partial_x^2+(-i\Sigma_x-i\alpha\Sigma_y\Sigma+u_yI-\alpha^3u_x\Sigma)\partial_x.\qquad (3b)$

Here

$\Sigma=\sum_{j=1}^3S_j\sigma_j,\qquad (4)$

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.