Синус-Гордон теңдеуі: Гиперболикалық дербес дифференциал теңдеулер

Hyperbolic partial differential equation
The sine Gordon equation is a nonlinear hyperbolic partial differential equation for a function dependent on two variables typically denoted and , involving the wave operator and the sine of
It was originally introduced by in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3 dimensional space. The equation was rediscovered by in their study of crystal dislocations known as the Frenkel–Kontorova model. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions, and is an example of an integrable PDE. Among well known integrable PDEs, the sine Gordon equation is the only relativistic system due to its Lorentz invariance.

The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by Bianchi and Bäcklund led to the discovery of Bäcklund transformations. Another transformation of pseudospherical surfaces is the Lie transform introduced by Sophus Lie in 1879, which corresponds to Lorentz boosts for solutions of the sine Gordon equation. There are also some more straightforward ways to construct new solutions but which do not give new surfaces. Since the sine Gordon equation is odd, the negative of any solution is another solution. However this does not give a new surface, as the sign change comes down to a choice of direction for the normal to the surface. New solutions can be found by translating the solution: if is a solution, then so is for an integer.

Consider a line of pendula, hanging on a straight line, in constant gravity. Connect the bobs of the pendula together by a string in constant tension. Let the angle of the pendulum at location be , then schematically, the dynamics of the line of pendulum follows Newton's second law:and this is the sine Gordon equation, after scaling time and distance appropriately. Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely , but more accurately However this does give an intuitive picture for the sine gordon equation. One can produce exact mechanical realizations of the sine gordon equation by more complex methods.

For different values of the parameter , the renormalizability properties of the sine Gordon theory change. The identification of these regimes is attributed to Jürg Fröhlich. The finite regime is , where no counterterms are needed to render the theory well posed. The super renormalizable regime is , where a finite number of counterterms are needed to render the theory well posed. More counterterms are needed for each threshold passed. For , the theory becomes ill defined The boundary values are and , which are respectively the free fermion point, as the theory is dual to a free fermion via the Coleman correspondence, and the self dual point, where the vertex operators form an affine sl2 subalgebra, and the theory becomes strictly renormalizable (renormalizable, but not super renormalizable).

A supersymmetric extension of the sine Gordon model also exists. Integrability preserving boundary conditions for this extension can be found as well. The sine Gordon model is in the same universality class as the effective action for a Coulomb gas of vortices and anti vortices in the continuous classical XY model, which is a model of magnetism. The Kosterlitz–Thouless transition for vortices can therefore be derived from a renormalization group analysis of the sine Gordon field theory. The sine Gordon equation also arises as the formal continuum limit of a different model of magnetism, the quantum Heisenberg model, in particular the XXZ model.