Ishimori equation

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

Equation

The Ishimori equation has the form Template:NumBlk Template:NumBlk

Lax representation

The Lax representation Template:NumBlk of the equation is given by Template:NumBlk Template:NumBlk

Here Template:NumBlk 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.

See also

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

References

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External links

• Ishimori_system at the dispersive equations wiki

Template:CMP-stub Template:Electromagnetism-stub