Rahul Sharma (Editor)

Ishimori equation

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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).

Contents

Equation

The Ishimori Equation has the form

∂ S ∂ t = S ∧ ( ∂ 2 S ∂ x 2 + ∂ 2 S ∂ y 2 ) + ∂ u ∂ x ∂ S ∂ y + ∂ u ∂ y ∂ S ∂ x , ( 1 a ) ∂ 2 u ∂ x 2 − α 2 ∂ 2 u ∂ y 2 = − 2 α 2 S ⋅ ( ∂ S ∂ x ∧ ∂ S ∂ y ) . ( 1 b )

Lax representation

The Lax representation

L t = A L − L A ( 2 )

of the equation is given by

L = Σ ∂ x + α I ∂ y , ( 3 a ) A = − 2 i Σ ∂ x 2 + ( − i Σ x − i α Σ y Σ + u y I − α 3 u x Σ ) ∂ x . ( 3 b )

Here

Σ = ∑ j = 1 3 S j σ j , ( 4 )

the σ i are the Pauli matrices and I is the identity matrix.

Reductions

IE 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 IE is the Davey-Stewartson equation.

References

Ishimori equation Wikipedia
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