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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).

Contents

EquationEdit

The Ishimori equation has the form

 
 

Lax representationEdit

The Lax representation

 

of the equation is given by

 
 

Here

 

the   are the Pauli matrices and   is the identity matrix.

ReductionsEdit

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

ReferencesEdit

  • 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 linksEdit