Ishimori equation

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

Equation

The IE has the form

:$frac\{partial\; mathbf\{S\{partial\; t\}\; =\; mathbf\{S\}wedge\; left(frac\{partial^2\; mathbf\{S\{partial\; x^\{2\; +\; frac\{partial^2\; mathbf\{S\{partial\; y^\{2\; ight)+\; frac\{partial\; u\}\{partial\; x\}frac\{partial\; mathbf\{S\{partial\; y\}\; +\; frac\{partial\; u\}\{partial\; y\}frac\{partial\; mathbf\{S\{partial\; x\},qquad\; (1a)$

:$frac\{partial^2\; u\}\{partial\; x^2\}-alpha^2\; frac\{partial^2\; u\}\{partial\; y^2\}=-2alpha^2\; mathbf\{S\}cdotleft(frac\{partial\; mathbf\{S\{partial\; x\}wedge\; frac\{partial\; mathbf\{S\{partial\; y\}\; ight).qquad\; (1b)$

Lax representation

The Lax representation

:$L\_t=AL-LAqquad\; (2)$

of the equation is given by

:$L=Sigma\; partial\_x+alpha\; Ipartial\_y,qquad\; (3a)$

:$A=\; -2iSigmapartial\_x^2+(-iSigma\_x-ialphaSigma\_ySigma+u\_yI-alpha^3u\_xSigma)partial\_x.qquad\; (3b)$

Here

:$Sigma=sum\_\{j=1\}^3S\_jsigma\_j,qquad\; (4)$

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.

ee also

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

References

*
*citation|last=Ishimori|first=Y.|title=Multi-vortex solutions of a two-dimensional nonlinear wave equation|journal=Prog. Theor. Phys. |volume=72|year=1984|pages=33-37|id=MR|0760959
doi= 10.1143/PTP.72.33
*citation|last=Konopelchenko |first=B.G. |title=Solitons in multidimensions|publisher= World Scientific |year=1993|isbn=978-9810213480
*citation|last= Martina|first= 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
*citation|editor1-first=David H.|editor1-last= Sattinger|editor2-first= C. A. |editor2-last=Tracy |title=Inverse Scattering and Applications|year=1991
id=MR|1135850
editor3-first=S.|editor3-last= Venakides|series= Contemporary Mathematics|volume= 122|publisher= American Mathematical Society|place= Providence, RI|year= 1991| ISBN:=0-8218-5129-2
*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

External links

* [http://tosio.math.toronto.edu/wiki/index.php/Ishimori_system Ishimori_system] at the dispersive equations wiki