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Ishimori equation

The Ishimori equation (IE) is a partial differential equation proposed by the Japanese mathematician Yuji 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

<math> \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)</math>
<math> \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)</math>

Lax representation

The Lax representation

<math>L_t=AL-LA\qquad (2)</math>

of the equation is given by

<math>L=\Sigma \partial_x+\alpha I\partial_y,\qquad (3a)</math>
<math>A= -2i\Sigma\partial_x^2+(-i\Sigma_x-i\alpha\Sigma_y\Sigma+u_yI-\alpha^3u_x\Sigma)\partial_x.\qquad (3b)</math>

Here

<math>\Sigma=\sum_{j=1}^3S_j\sigma_j,\qquad (4)</math>

the <math>\sigma_i</math> are the Pauli matrices and <math>I</math> 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.

See also

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, doi:10.1134/1.1648299 
  • Ishimori, Yuji (1984), "Multi-vortex solutions of a two-dimensional nonlinear wave equation", Prog. Theor. Phys. 72: 33–37, MR 0760959, doi:10.1143/PTP.72.33 
  • 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, 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, 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 

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