Sine-Gordon equation
[1] The equation was rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model.
[2] This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions,[3] 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.
Passing to the light-cone coordinates (u, v), akin to asymptotic coordinates where the equation takes the form[5] This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of surfaces of constant Gaussian curvature K = −1, also called pseudospherical surfaces.
Conversely, one can start with a solution to the sine-Gordon equation to obtain a pseudosphere uniquely up to rigid transformations.
There is a theorem, sometimes called the fundamental theorem of surfaces, that if a pair of matrix-valued bilinear forms satisfy the Gauss–Codazzi equations, then they are the first and second fundamental forms of an embedded surface in 3-dimensional space.
Solutions to the sine-Gordon equation can be used to construct such matrices by using the forms obtained above.
Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely
One can produce exact mechanical realizations of the sine-gordon equation by more complex methods.
The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model introduced by Julio Rubinstein in 1970.
[8] Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge
[10] The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons.
The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a phase shift.
Since the colliding solitons recover their velocity and shape, such an interaction is called an elastic collision.
Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a breather.
is a solution of the sine-Gordon equation Then the system where a is an arbitrary parameter, is solvable for a function
By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation.
With it the first two terms in the Taylor expansion of the potential coincide with the potential of a massive scalar field, as mentioned in the naming section; the higher order terms can be thought of as interactions.
The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined
[16] It is possible to find boundary conditions which preserve the integrability of the model.
[16] On a half line the spectrum contains boundary bound states in addition to the solitons and breathers.
[16] In quantum field theory the sine-Gordon model contains a parameter that can be identified with the Planck constant.
The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of breathers.
Semi-classical quantization of the sine-Gordon model was done by Ludwig Faddeev and Vladimir Korepin.
[20] The exact quantum scattering matrix was discovered by Alexander Zamolodchikov.
This article also showed that the constants appearing in the model behave nicely under renormalization: there are three parameters
The quantum sine-Gordon equation should be modified so the exponentials become vertex operators with
, 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).
The stochastic or dynamical sine-Gordon model has been studied by Martin Hairer and Hao Shen [25] allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting.
[28][29] The Kosterlitz–Thouless transition for vortices can therefore be derived from a renormalization group analysis of the sine-Gordon field theory.
