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 |
Equation
The IE has the form
Lax representation
of the equation is given by
Here
the σi are the Pauli matrices and I 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
- Nonlinear Schrödinger equation
- Heisenberg model (classical)
- Spin wave
- Landau-Lifshitz equation
- 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:
- Ishimori, Y. (1984), "Multi-vortex solutions of a two-dimensional nonlinear wave equation", Prog. Theor. Phys. 72: 33–37, doi:, MR0760959
- 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, doi:
- Sattinger, David H.; Tracy, C. A.; Venakides, S., eds. (1991), Inverse Scattering and Applications, Contemporary Mathematics, 122, Providence, RI: American Mathematical Society, MR1135850, ISBN 0821851292
- Sung, Li-yeng (1996), "The Cauchy problem for the Ishimori equation", Journal of Functional Analysis 139: 29–67, doi:
External links
- Ishimori_system at the dispersive equations wiki
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
