Ishimori equation

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).

Equation

The Ishimori equation has the form

[math]\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}, }[/math]

(1a)

[math]\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). }[/math]

(1b)

Lax representation

The Lax representation

[math]\displaystyle{ L_t = AL - LA }[/math]

(2)

of the equation is given by

[math]\displaystyle{ L =\Sigma \partial_x + \alpha I\partial_y, }[/math]

(3a)

[math]\displaystyle{ A = -2i\Sigma\partial_x^2+(-i\Sigma_x-i\alpha\Sigma_y\Sigma+u_yI-\alpha^3u_x\Sigma)\partial_x. }[/math]

(3b)

Here

[math]\displaystyle{ \Sigma=\sum_{j=1}^3S_j\sigma_j, }[/math]

(4)

the [math]\displaystyle{ \sigma_i }[/math] are the Pauli matrices and [math]\displaystyle{ I }[/math] 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.

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, Bibcode: 2003JETPL..78..740G
Ishimori, Yuji (1984), "Multi-vortex solutions of a two-dimensional nonlinear wave equation", Prog. Theor. Phys. 72 (1): 33–37, doi:10.1143/PTP.72.33, Bibcode: 1984PThPh..72...33I
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, PMID 10010201, Bibcode: 1994PhRvB..4912915M
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, https://archive.org/details/inversescatterin0000amsi
Sung, Li-yeng (1996), "The Cauchy problem for the Ishimori equation", Journal of Functional Analysis 139: 29–67, doi:10.1006/jfan.1996.0078

External links

Ishimori_system at the dispersive equations wiki

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Ishimori equation. Read more