The Ishimori equation 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
|
(1a)
|
|
(1b)
|
|
(2)
|
of the equation is given by
|
(3a)
|
|
(3b)
|
Here
|
(4)
|
the are the Pauli matrices and is the identity matrix.
The Ishimori equation admits an important reduction: in 1+1 dimensions it reduces to the continuous classical Heisenberg ferromagnet equation (CCHFE). The CCHFE is integrable.
The equivalent counterpart of the Ishimori equation is the Davey-Stewartson equation.