Ishimori equation

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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 representation[edit]

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 counterpart[edit]

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

See also[edit]


  • 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 links[edit]