New Travelling Wave Solutions for Sine-Gordon Equation

We propose a method to deal with the general sine-Gordon equation. Several new exact travelling wave solutions with the form of JacobiAmplitude function are derived for the general sine-Gordon equation by using some reasonable transformation. Compared with previous solutions, our solutions are more general than some of the previous

Towards soliton solutions of a perturbed sine-Gordon equation

We give arguments for the existence of {\it exact} travelling-wave (in particular solitonic) solutions of a perturbed sine-Gordon equation on the real line or on the circle, and classify them. The perturbation of the equation consists of a constant forcing term and a linear dissipative term. Such solutions are allowed exactly by the energy balance of these terms, and can be observed experimentally e.g. in the Josephson effect in the theory of superconductors, which is one of the physical phenomena described by the equation.Comment: 16 pages, 4 figures include

Solitary waves and nonlinear Klein-Gordon Equations

We analytically study the kink-antikink (K-K) collisions in the classical one spatial dimension and time phi-fourth field theory as an example of inelastic collisions between solitary waves. We use the linear eigenvalue collective coordinate approach to describe the system in terms of the separation distance between the kink and the antikink and the amplitude of shape vibrations generated on each kink as a result of the collision. By calculating the energy given to the shape vibrations as a function of the incoming velocity, we find the critical value of the initial velocity above which the two colliding kinks always separate after the collision. A model previously proposed to explain the two-bounce collisions in terms of a resonant energy exchange between the orbital frequency of the bound K-K pair and the frequency of shape vibrations is modified using our analytical results. We derive a (data-free) formula that predicts the values of the initial velocities for which resonance occurs. A generalized version of this modified model is shown to give good results when it is applied to K-K collisions in other similar field theories. In the Appendices Nonlinear Klein Gordon equations with solitary (travelling) wave solutions are reviewed and solved for particular cases. The solutions are related to soliton solutions of the sine-Gordon equation. Also the phi-fourth equation perturbed with a constant force and dissipation is solved, and finally, we present new kink-bearing integro-differential and nonlinear differential equations

On kinks and other travelling-wave solutions of a modified sine-Gordon equation

We give an exhaustive, non-perturbative classification of exact travelling-wave solutions of a perturbed sine-Gordon equation (on the real line or on the circle) which is used to describe the Josephson effect in the theory of superconductors and other remarkable physical phenomena. The perturbation of the equation consists of a constant forcing term and a linear dissipative term. On the real line candidate orbitally stable solutions with bounded energy density are either the constant one, or of kink (i.e. soliton) type, or of array-of-kinks type, or of "half-array-of-kinks" type. While the first three have unperturbed analogs, the last type is essentially new. We also propose a convergent method of successive approximations of the (anti)kink solution based on a careful application of the fixed point theorem.Comment: Latex file, 25 pages, 4 figures. Final version to appear in "Meccanica

Sine-Gordon solitons, auxiliary fields, and singular limit of a double pendulums chain

We consider the continuum version of an elastic chain supporting topological and non-topological degrees of freedom; this generalizes a model for the dynamics of DNA recently proposed and investigated by ourselves. In a certain limit, the non-topological degrees of freedom are frozen, and the model reduces to the sine-Gordon equations and thus supports well-known topological soliton solutions. We consider a (singular) perturbative expansion around this limit and study in particular how the non-topological field assume the role of an auxiliary field. This provides a more general framework for the slaving of this degree of freedom on the topological one, already observed elsewhere in the context of the mentioned DNA model; in this framework one expects such phenomenon to arise in a quite large class of field-theoretical models.Comment: 18 pages, 2 figure