Ishimori equation

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).

Equation

The Ishimori equation has the form

$$\frac{\partial 𝐒}{\partial t}=𝐒\wedge (\frac{{\partial }^2𝐒}{\partial x^2}+\frac{{\partial }^2𝐒}{\partial y^2})+\frac{\partial u}{\partial x}\frac{\partial 𝐒}{\partial y}+\frac{\partial u}{\partial y}\frac{\partial 𝐒}{\partial x},$$ 1a

$$\frac{{\partial }^{2}u}{\partial x^2}-{\alpha }^2\frac{{\partial }^{2}u}{\partial y^2}=-2{\alpha }^2𝐒\cdot (\frac{\partial 𝐒}{\partial x}\wedge \frac{\partial 𝐒}{\partial y}).$$ 1b

Lax representation

The Lax representation

$$L_t=AL-LA$$ 2

of the equation is given by

$$L=\Sigma {\partial }_x+\alpha I{\partial }_y,$$ 3a

$$A=-2i\Sigma \underset{x}{\overset{2}{\partial }}+(-i{\Sigma }_x-i\alpha {\Sigma }_y\Sigma +u_{y}I-{\alpha }^3{u}_x\Sigma ){\partial }_x.$$ 3b

Here

$$\Sigma =\sum _{j=1}^3{S}_j{\sigma }_j,$$ 4

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

External links

• Ishimori_system at the dispersive equations wiki