THC Science

The Ishimori equation 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}},

(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).

(1b)

L_{t}=AL-LA

(2)

of the equation is given by

L=\Sigma \partial _{x}+\alpha I\partial _{y},

(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}.

(3b)

Here

\Sigma =\sum _{j=1}^{3}S_{j}\sigma _{j},

(4)

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

The Ishimori equation admits an important reduction: in 1+1 dimensions it reduces to the continuous classical Heisenberg ferromagnet equation (CCHFE). The CCHFE is integrable.

The equivalent counterpart of the Ishimori equation is the Davey-Stewartson equation.

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Ishimori, Yuji (1984), "Multi-vortex solutions of a two-dimensional nonlinear wave equation", Prog. Theor. Phys., 72 (1): 33–37, Bibcode:1984PThPh..72...33I, doi:10.1143/PTP.72.33, MR 0760959
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Sung, Li-yeng (1996), "The Cauchy problem for the Ishimori equation", Journal of Functional Analysis, 139: 29–67, doi:10.1006/jfan.1996.0078