Ishimori equation

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

Contents

[edit] Equation

The IE 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},\qquad (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).\qquad (1b)

[edit] 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_{j=1}^3S_j\sigma_j,\qquad (4)

the σi are the Pauli matrices and I is the identity matrix.

[edit] Reductions

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

[edit] Equivalent counterpart

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

[edit] See also

[edit] References

  • Gutshabash, E.Sh. (2003), “Generalized Darboux transform in the Ishimori magnet model on the backround of spiral structures”, JETP Letters 78 (11): 740-744 
  • Ishimori, Y. (1984), “Multi-vortex solutions of a two-dimensional nonlinear wave equation”, Prog. Theor. Phys. 72: 33-37, MR0760959, DOI 10.1143/PTP.72.33 
  • Konopelchenko, B.G. (1993), Solitons in multidimensions, World Scientific, ISBN 978-9810213480 
  • 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: 12915 - 12922 
  • Sattinger, David H.; Tracy, C. A. & Venakides, S., eds. (1991), Inverse Scattering and Applications, vol. 122, Contemporary Mathematics, Providence, RI: American Mathematical Society, MR1135850 
  • Sung, Li-yeng (1996), “The Cauchy problem for the Ishimori equation”, Journal of Functional Analysis 139: 29-67, DOI 10.1006/jfan.1996.0078 

[edit] External links