The Ishimori equation is a partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

 proposed by the Japanese mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable

Integrable system

In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...

 .

Equation

The IE has the form

Lax representation

The Lax representation


of the equation is given by



Here


the are the Pauli matrices

Pauli matrices

The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

 and is the identity matrix.

Reductions

IE admits an important reduction:
in 1+1 dimensions it reduces to the continuous classical Heisenberg ferromagnet equation

Heisenberg model (classical)

The Classical Heisenberg model is the n = 3 case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena.-Definition:...

 (CCHFE). The CCHFE is integrable.

Equivalent counterpart

The equivalent counterpart of the IE is the Davey-Stewartson equation.

See also

• Nonlinear Schrödinger equation
  Nonlinear Schrödinger equation
  In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...
  
  
• Heisenberg model (classical)
  Heisenberg model (classical)
  The Classical Heisenberg model is the n = 3 case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena.-Definition:...
  
  
• Spin wave
  Spin wave
  Spin waves are propagating disturbances in the ordering of magnetic materials. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are boson modes of the spin lattice...
  
  
• Landau-Lifshitz equation
• Soliton
  Soliton
  In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
  
  
• Vortex
  Vortex
  A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
  
  
• Nonlinear systems
• Davey–Stewartson equation
  Davey–Stewartson equation
  In fluid dynamics, the Davey–Stewartson equation was introduced in a paper by to describe the evolution of a three-dimensional wave-packet on water of finite depth....
  
  

External links

• Ishimori_system at the dispersive equations wiki