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

The Lax representation

of the equation is given by

Here

the ${\displaystyle \sigma _{i}}$ are the Pauli matrices and ${\displaystyle I}$ 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.

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

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