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

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

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

Lax representation

The Lax representation

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

of the equation is given by

${\displaystyle L=\Sigma \partial _{x}+\alpha I\partial _{y},\qquad (3a)}$

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

Here

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

the ${\displaystyle \sigma _{i))$ are the Pauli matrices and ${\displaystyle I}$ 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.