View source for Ishimori equation

The '''Ishimori equation''' is a [[partial differential equation]] proposed by the Japanese [[mathematician]] {{harvtxt|Ishimori|1984}}. Its interest is as the first example of a nonlinear spin-one field model in the plane that is [[integrable system|integrable]] {{harv|Sattinger|Tracy|Venakides|1991|p=78}}.

==Equation==
The Ishimori equation has the form
{{NumBlk||<math display="block"> \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},</math>|{{EquationRef|1a}}}}
{{NumBlk||<math display="block"> \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).</math>|{{EquationRef|1b}}}}

==Lax representation==

The [[Lax representation]]
{{NumBlk||<math display="block">L_t = AL - LA</math>|{{EquationRef|2}}}}
of the equation is given by
{{NumBlk||<math display="block">L =\Sigma \partial_x + \alpha I\partial_y,</math>|{{EquationRef|3a}}}}
{{NumBlk||<math display="block">A = -2i\Sigma\partial_x^2+(-i\Sigma_x-i\alpha\Sigma_y\Sigma+u_yI-\alpha^3u_x\Sigma)\partial_x.</math>|{{EquationRef|3b}}}}

Here
{{NumBlk||<math display="block">\Sigma=\sum_{j=1}^3S_j\sigma_j,</math>|{{EquationRef|4}}}}
the <math>\sigma_i</math> are the [[Pauli matrices]] and <math>I</math> is the identity matrix.

==Reductions==
The Ishimori equation admits an important reduction:
in 1+1 dimensions it reduces to the [[Heisenberg model (classical)|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==
* {{citation|last=Gutshabash |first=E.Sh. |title=Generalized Darboux transform in the Ishimori magnet model on the background of spiral structures|journal=JETP Letters|volume=78|issue=11|pages=740–744 |year=2003|doi=10.1134/1.1648299|arxiv=nlin/0409001|bibcode=2003JETPL..78..740G|s2cid=16905805 }}
*{{citation|last=Ishimori|first=Yuji|title=Multi-vortex solutions of a two-dimensional nonlinear wave equation|journal=Prog. Theor. Phys. |volume=72|year=1984|issue=1 |pages=33–37
|doi= 10.1143/PTP.72.33|mr=0760959|bibcode=1984PThPh..72...33I|doi-access=free}}
*{{citation|last=Konopelchenko |first=B.G. |title=Solitons in multidimensions|publisher= World Scientific |year=1993|isbn=978-981-02-1348-0}}
*{{citation|last1= Martina|first1= L.|last2= Profilo |first2=G.|last3= Soliani|first3= G.|last4= Solombrino |first4=L. |title=Nonlinear excitations in a Hamiltonian spin-field model in 2+1 dimensions |journal=Phys. Rev. B |volume=49|pages=12915–12922 |year=1994|doi= 10.1103/PhysRevB.49.12915|issue= 18|pmid= 10010201|bibcode=1994PhRvB..4912915M}}
*{{citation|editor1-first=David H.|editor1-last=Sattinger|editor2-first=C. A.|editor2-last=Tracy|title=Inverse Scattering and Applications|year=1991|editor3-first=S.|editor3-last=Venakides|series=Contemporary Mathematics|volume=122|publisher=American Mathematical Society|place=Providence, RI|isbn=0-8218-5129-2|mr=1135850|doi=10.1090/conm/122|url-access=registration|url=https://archive.org/details/inversescatterin0000amsi}}
*{{citation|first=Li-yeng |last=Sung|title= The Cauchy problem for the Ishimori equation|journal= Journal of Functional Analysis|volume=139|pages=29–67 |year=1996|doi=10.1006/jfan.1996.0078|doi-access=free}}
==External links==
*[https://web.archive.org/web/20080820111240/http://tosio.math.toronto.edu/wiki/index.php/Ishimori_system Ishimori_system] at the dispersive equations wiki

[[Category:Electric and magnetic fields in matter]]
[[Category:Partial differential equations]]
[[Category:Integrable systems]]


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