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 & Venakides 1991, p. 78).

Equation

The Ishimori equation has the form

{{{2}}}

 

 

 

 

({{{3}}})

{\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}}

{{{2}}}

 

 

 

 

({{{3}}})

{\partial x}\wedge \frac{\partial \mathbf{S}}{\partial y}\right).</math>|1b}}

Lax representation

The Lax representation

2

 

 

 

 

({{{3}}})

of the equation is given by

3a

 

 

 

 

({{{3}}})

3b

 

 

 

 

({{{3}}})

Here

4

 

 

 

 

({{{3}}})

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

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, arXiv:nlin/0409001, Bibcode:2003JETPL..78..740G, doi:10.1134/1.1648299, S2CID 16905805
• Ishimori, Yuji (1984), "Multi-vortex solutions of a two-dimensional nonlinear wave equation", Prog. Theor. Phys., 72 (1): 33–37, Bibcode:1984PThPh..72...33I, doi:10.1143/PTP.72.33, MR 0760959
• 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, Bibcode:1994PhRvB..4912915M, doi:10.1103/PhysRevB.49.12915, PMID 10010201
• Sattinger, David H.; Tracy, C. A.; Venakides, S., eds. (1991), Inverse Scattering and Applications, Contemporary Mathematics, vol. 122, Providence, RI: American Mathematical Society, doi:10.1090/conm/122, 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

External links

• Ishimori_system at the dispersive equations wiki