Ishimori Equation
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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

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

## 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.

## References

• Gutshabash, E.Sh. (2003), "Generalized Darboux transform in the Ishimori magnet model on the background of spiral structures", JETP Letters, 78 (11): 740-744, arXiv:nlin/0409001, Bibcode:2003JETPL..78..740G, doi:10.1134/1.1648299
• Ishimori, Yuji (1984), "Multi-vortex solutions of a two-dimensional nonlinear wave equation", Prog. Theor. Phys., 72: 33-37, Bibcode:1984PThPh..72...33I, doi:10.1143/PTP.72.33, MR 0760959
• Konopelchenko, B.G. (1993), Solitons in multidimensions, World Scientific, ISBN 978-981-02-1348-0
• Martina, L.; Profilo, G.; Soliani, G.; Solombrino, L. (1994), "Nonlinear excitations in a Hamiltonian spin-field model in 2+1 dimensions", Phys. Rev. B, 49 (18): 12915-12922, Bibcode:1994PhRvB..4912915M, doi:10.1103/PhysRevB.49.12915
• Sattinger, David H.; Tracy, C. A.; Venakides, S., eds. (1991), Inverse Scattering and Applications, Contemporary Mathematics, 122, Providence, RI: American Mathematical Society, doi:10.1090/conm/122, ISBN 0-8218-5129-2, MR 1135850
• Sung, Li-yeng (1996), "The Cauchy problem for the Ishimori equation", Journal of Functional Analysis, 139: 29-67, doi:10.1006/jfan.1996.0078