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).
Equation
The Ishimori equation has the form
\( {\displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \left({\frac {\partial ^{2}\mathbf {S} }{\partial x^{2}}}+{\frac {\partial ^{2}\mathbf {S} }{\partial y^{2}}}\right)+{\frac {\partial u}{\partial x}}{\frac {\partial \mathbf {S} }{\partial y}}+{\frac {\partial u}{\partial y}}{\frac {\partial \mathbf {S} }{\partial x}},\qquad (1a)} \)
\( {\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}-\alpha ^{2}{\frac {\partial ^{2}u}{\partial y^{2}}}=-2\alpha ^{2}\mathbf {S} \cdot \left({\frac {\partial \mathbf {S} }{\partial x}}\wedge {\frac {\partial \mathbf {S} }{\partial y}}\right).\qquad (1b)} \)
Lax representation
The Lax representation
\( {\displaystyle L_{t}=AL-LA\qquad (2)} \)
of the equation is given by
\( {\displaystyle L=\Sigma \partial _{x}+\alpha I\partial _{y},\qquad (3a)} \)
\( {\displaystyle A=-2i\Sigma \partial _{x}^{2}+(-i\Sigma _{x}-i\alpha \Sigma _{y}\Sigma +u_{y}I-\alpha ^{3}u_{x}\Sigma )\partial _{x}.\qquad (3b)} \)
Here
\( {\displaystyle \Sigma =\sum _{j=1}^{3}S_{j}\sigma _{j},\qquad (4)} \)
the \( \sigma _{i} \) 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.
See also
Nonlinear Schrödinger equation
Heisenberg model (classical)
Spin wave
Landau–Lifshitz model
Soliton
Vortex
Nonlinear systems
Davey–Stewartson equation
References
Gutshabash, E.Sh. (2003), "Generalized Darboux transform in the Ishimori magnet model on the background of spiral structures", JETP Letters, 78 (11): 740–744, arXiv:nlin/0409001, Bibcode:2003JETPL..78..740G, doi:10.1134/1.1648299
Ishimori, Yuji (1984), "Multi-vortex solutions of a two-dimensional nonlinear wave equation", Prog. Theor. Phys., 72: 33–37, Bibcode:1984PThPh..72...33I, doi:10.1143/PTP.72.33, MR 0760959
Konopelchenko, B.G. (1993), Solitons in multidimensions, World Scientific, ISBN 978-981-02-1348-0
Martina, L.; Profilo, G.; Soliani, G.; Solombrino, L. (1994), "Nonlinear excitations in a Hamiltonian spin-field model in 2+1 dimensions", Phys. Rev. B, 49 (18): 12915–12922, Bibcode:1994PhRvB..4912915M, doi:10.1103/PhysRevB.49.12915
Sattinger, David H.; Tracy, C. A.; Venakides, S., eds. (1991), Inverse Scattering and Applications, Contemporary Mathematics, 122, Providence, RI: American Mathematical Society, doi:10.1090/conm/122, ISBN 0-8218-5129-2, MR 1135850
Sung, Li-yeng (1996), "The Cauchy problem for the Ishimori equation", Journal of Functional Analysis, 139: 29–67, doi:10.1006/jfan.1996.0078
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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