Ishimori equation

The Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .

Equation

The Ishimori equation has the form

:$\frac = \mathbf\wedge \left(\frac + \frac\right)+ \frac\frac + \frac\frac,\qquad (1a)$

:$\frac-\alpha^2 \frac=-2\alpha^2 \mathbf\cdot\left(\frac\wedge \frac\right).\qquad (1b)$

Lax representation

The 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

:$\Sigma=\sum_^3S_j\sigma_j,\qquad (4)$

the $\sigma_i$ are the Pauli matrices and $I$ is the identity matrix.

Reductions

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.

Equivalent counterpart

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

See also

* Nonlinear Schrödinger equation
* Heisenberg model (classical)
* Spin wave
* Landau–Lifshitz model
* Soliton
* Vortex
* Nonlinear systems
* Davey–Stewartson equation

References

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External links

Ishimori_system
at the dispersive equations wiki

Electric and magnetic fields in matter
Partial differential equations
Integrable systems

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