# Ishimori equation

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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

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

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.

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## References

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