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The Ishimori equation is a
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
proposed by the Japanese
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
. Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .


Equation

The Ishimori equation has the form


Lax representation

The Lax representation of the equation is given by Here the \sigma_i are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
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 In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonli ...
*
Heisenberg model (classical) In statistical physics, the classical Heisenberg model, developed by Werner Heisenberg, is the n = 3 case of the n-vector model, ''n''-vector model, one of the models used to model ferromagnetism and other phenomena. Definition The classical Heise ...
* Spin wave * Landau–Lifshitz model *
Soliton In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
*
Vortex In fluid dynamics, a vortex (: vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
*
Nonlinear systems In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathem ...
* Davey–Stewartson equation


References

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External links


Ishimori_system
at the dispersive equations wiki Electric and magnetic fields in matter Partial differential equations Integrable systems {{electromagnetism-stub