Classical sine-Gordon Equation Research Articles

Some quantum effects in the sine-Gordon model under the gaussian approximation

By means of the gaussian approximation, the classical sine-Gordon equation evolves into a new equation. After rescaling space and time, the soliton and breather solutions of the new equation can be solved approximately and some elliptic function solutions can be obtained exactly. Some other quantum effects are also discussed, especially, a free massive fermion in 1 + 1 dimensional space corresponds exactly to a → kink of the quantum sG system rather than a 2→ kink.

Solution of the initial value problem for the sine-Gordon equation using a Kac-Moody algebra

We solve the classical sine-Gordon equation using a Lax pair belonging to a Kac-Moody algebra. By realising the algebra in terms of fermionic currents we reduce the initial value problem to the evaluation of a fermionic propagator as a sum of Feynman diagrams.

Comment on "New exact solutions of the classical sine-Gordon equation in 2+1 and 3+1 dimensions"

A critical discussion of Leibbrandt's recent solutions of the sine-Gordon equation in higher dimensions is given. Also it is shown that his transformation is not a Bäcklund transformation.Received 30 May 1980DOI:https://doi.org/10.1103/PhysRevD.23.2482©1981 American Physical Society

Reply to "Comment on `New exact solutions of the classical sine-Gordon equation in 2+1 and 3+1 dimensions"'

It is argued that a nonlinear transformation for the higher-dimensional sine-Gordon system is sufficiently powerful to generate, in conjunction with the appropriate constraint equations, exact multiple, albeit coplanar, solutions. The evidence in favor of multisoliton solutions is supported by the results from detailed numerical work, including the use of quadruple precision, the application of three-dimensional computer plotting routines, and the existence of a four-soliton solution of topological charge $Q=+4$.

New Exact Solutions of the Classical Sine-Gordon Equation in 2 + 1 and 3 + 1 Dimensions

The method of B\"acklund transformations is employed to derive in 2 + 1 and 3 + 1 dimensions exact solutions of the sine-Gordon equation $[{\ensuremath{\nabla}}^{2}\ensuremath{-}{c}^{\ensuremath{-}2}(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}})]\ensuremath{\chi}=sin\ensuremath{\chi}$. A formula is developed in 3 + 1 dimensions which permits us to generate without additional quadratures an infinite class of new time-dependent solutions.

Anomalies of currents in the quantized sine-gordon equation

We investigate the meaning of the (infinitely many) conserved non-trivial currents of the classical Sine-Gordon equation for conventional quantum perturbation theory of a scalar field φ 0000with selfinteraction (cos( Bφ) − 1 + ( ßφ) 2/2). Radiation corrections produce for all currents anomalies with contributions on the mass shell.