Ishimori equation

The Ishimori equation (IE) 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).


The Ishimori equation has the form


Lax representationEdit

The Lax representation


of the equation is given by




the   are the Pauli matrices and   is the identity matrix.


IE admits an important reduction: in 1+1 dimensions it reduces to the continuous classical Heisenberg ferromagnet equation (CCHFE). The CCHFE is integrable.

Equivalent counterpartEdit

The equivalent counterpart of the IE is the Davey-Stewartson equation.

See alsoEdit


  • 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
  • Ishimori, Yuji (1984), "Multi-vortex solutions of a two-dimensional nonlinear wave equation", Prog. Theor. Phys., 72: 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
  • 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, 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 linksEdit