Ishimori equation
The Ishimori equation (IE) is a partial differential equation proposed by the Japanese mathematician Yuji 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
Lax representation
The Lax representation
of the equation is given by
Here
the are the Pauli matrices and
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:10.1134/1.1648299
- Ishimori, Yuji (1984), "Multi-vortex solutions of a two-dimensional nonlinear wave equation", Prog. Theor. Phys. 72: 33–37, 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, doi:10.1103/PhysRevB.49.12915
- Sattinger, David H.; Tracy, C. A.; Venakides, S., eds. (1991), Inverse Scattering and Applications, Contemporary Mathematics 122, Providence, RI: American Mathematical Society, 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