Ishimori equation

The Ishimori equation is a partial differential equation proposed by the Japanese mathematician. Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable.

Equation
The Ishimori equation has the form

Lax representation
The Lax representation

of the equation is given by

Here

the $$\sigma_i$$ are the Pauli matrices and $$I$$ is the identity matrix.

Reductions
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.

Equivalent counterpart
The equivalent counterpart of the Ishimori equation is the Davey-Stewartson equation.