Ishimori equation explained

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 representationof the equation is given by

Herethe

are the Pauli matrices and 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.

See also

• Nonlinear Schrödinger equation
• Heisenberg model (classical)
• Spin wave
• Landau–Lifshitz model
• Soliton
• Vortex
• Nonlinear systems
• Davey–Stewartson equation

External links

• Ishimori_system at the dispersive equations wiki