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
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
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
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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 ...
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Spin wave
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Landau–Lifshitz model
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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 ...
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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_systemat the dispersive equations wiki
Electric and magnetic fields in matter
Partial differential equations
Integrable systems
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