Physics: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

$$\frac{\partial \mathbf{𝐒}}{\partial t}=\mathbf{𝐒}\wedge (\frac{{\partial }^2\mathbf{𝐒}}{\partial x^2}+\frac{{\partial }^2\mathbf{𝐒}}{\partial y^2})+\frac{\partial u}{\partial x}\frac{\partial \mathbf{𝐒}}{\partial y}+\frac{\partial u}{\partial y}\frac{\partial \mathbf{𝐒}}{\partial x},$$

(1a)

$$\frac{{\partial }^{2}u}{\partial x^2}-{\alpha }^2\frac{{\partial }^{2}u}{\partial y^2}=-2{\alpha }^2\mathbf{𝐒}\cdot (\frac{\partial \mathbf{𝐒}}{\partial x}\wedge \frac{\partial \mathbf{𝐒}}{\partial y}).$$

(1b)

Lax representation

The Lax representation

$$L_t=AL-LA$$

(2)

of the equation is given by

$$L=\mathrm{\Sigma }{\partial }_x+\alpha I{\partial }_y,$$

(3a)

$$A=-2i\mathrm{\Sigma }\underset{x}{\overset{2}{\partial }}+(-i{\mathrm{\Sigma }}_x-i\alpha {\mathrm{\Sigma }}_y\mathrm{\Sigma }+u_{y}I-{\alpha }^3{u}_x\mathrm{\Sigma }){\partial }_x.$$

(3b)

Here

$$\mathrm{\Sigma }=\sum _{j=1}^3{S}_j{\sigma }_j,$$

(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

• 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