The sine-Gordon Model and its Applications

This chapter offers an overview of the vast research area developed around the sine-Gordon (sG) equation, including solution methods and various nonlinear modes generated by this equation, viz., topological and dynamical solitons (kinks and breathers), cnoidal waves (chains of kinks), and others. Also included is a survey of physical applications of the sG equation, and an outline of the perturbation theory used for the analysis of physical models based on the sG equation but differing from the ideal integrable form of this equation. Topics presented in the chapter, with some details or in a brief form, are the inverse scattering, transform and the perturbation theory based on it, the Bäcklund transform, energy- and momentum-balance methods for the analysis of the soliton dynamics in perturbed versions of the sG equation, the double sG equation, quantum sG systems, multidimensional sG equations, systems of coupled sG equations, and others.

In the present chapter, we consider two prototypical Klein–Gordon models: the integrable sine-Gordon equation and the non-integrable ϕ ⁴ model. We focus, in particular, on two of their principal solutions, namely the kink-like heteroclinic connections and the time-periodic, exponentially localized in space breather waveforms. Two limits of the discrete variants of these models are contrasted: on the one side, the analytically tractable original continuum limit, and on the opposite end, the highly discrete, so-called anti-continuum limit of vanishing coupling. Numerical computations are used to bridge these two limits, as regards the existence, stability and dynamical properties of the waves. Finally, a recent variant of this theme is presented in the form of \(\mathcal{P}\mathcal{T}\)-symmetric Klein–Gordon field theories and a number of relevant results are touched upon.

Soliton collisions constitute a fascinating topic of nonlinear science and its numerous applications. Solitary waves are well-known to be robust, being able to maintain their character upon mutual collisions and interactions with imperfections of the media. Therefore, they are efficient carriers of various physical quantities, e.g., energy, momentum, topological charge, etc. However, only in exactly integrable systems the soliton collisions are perfectly elastic. The addition of different types of perturbations to such models produces a number of intriguing effects such as radiation of small-amplitude wave packets, the possibility of excitation of the soliton internal modes, energy exchange between the solitons’ internal and translational modes, radiationless energy exchange between colliding solitons, to name only some of the principal ones. The aim of this chapter is to present a brief historical overview of the theme of soliton collisions and a description of its current state-of-the-art within the realm of perturbed sine-Gordon and related Klein–Gordon equations.

We present a modulation approach coupled with radiation to study the behavior of one and two space dimensional coherent structures in the sG equation. We show how the radiation induced motion can be coupled to the collective coordinates which describe the soliton. It is also shown how the usual geometric optics can be used to calculate the effect of two dimensional spiral waves on solitons. In turn this allows to study the stabilization of coherent solutions by internal degrees of freedom or the so-called PN effect. We point out some open questions in the last section.

The pendulum chain with torsional spring coupling represents a physical system that is described exactly by the spatially discrete sG equation. In order to study its solutions experimentally, however, we have to drive the system to counteract the inevitable dissipation we face in experiments. In fact, it is important to model the main sources of energy dissipation realistically; here we find that both on-site and intersite contributions to dissipation are relevant. We show that lattice solitons, also known as discrete breathers (DB) or intrinsic localized modes (ILMs), can be produced and stabilized in this system in the presence of driving (forcing) and damping. One way to do this is by exploiting the modulational instability of the uniform mode. Once a discrete soliton has been phase-locked to the driver, it persists indefinitely, and its properties can be studied, as can the range of its stability in parameter space and the types of instability outside of it. We also explore in some detail the interesting effect of which there exists no analogue in continuous media: the existence of two types of discrete solitons of different symmetry—the one-site centered and the two-site centered ILM. Furthermore, we find an exchange of stability between those two breather types as certain system parameters are varied. Comparison to analytical work based on realistic model parameters will be discussed as well.

This chapter is an overview of the soliton ratchets (motion of solitons under zero-average external forces) in the driven and damped sine-Gordon equation. Soliton ratchets induced by the breaking of spatial, temporal and field symmetries are discussed in detail. Analytical methods, such as collective coordinates theory and symmetry analysis, are presented, together with their comparison with numerical simulations. Symmetry analysis, based on the time-shift invariance of the average velocity of solitons, explains in an unified way, intrinsic phenomena of the ratchets such as: the control of reversal currents by means of the amplitudes and phases of the drivers, the motion induced by damping, and the suppression of the current when certain symmetries of the forces hold. Finally, it is pointed out that some of the previous results require verification through numerical simulations and experiments.

Superconducting Josephson-junctions are excellent experimental systems for the general study of nonlinear phenomena and nonlinear localised excitations. Specifically, a long Josephson-junction is described by the continuous sine-Gordon equation and a Josephson-junction parallel array by its discrete counterpart. This chapter constitute a revision of the physics of such superconducting systems in the light of the sine-Gordon equation.

We discuss in an expository style some results,old and new,regarding connections between solitons and black holes in 2d dilaton gravity, for both Lorentzian and Euclidean metrics. Black hole vacua (the ground state or mass zero case), and non sine-Gordon solitons such as dissipatons (solutions of a resonance nonlinear Schrödinger equation) are also considered.

Motivated by integrability of the sine-Gordon equation, we investigate a technique for constructing desired solutions to Einstein’s equations by combining a dressing technique with a control-theory approach. After reviewing classical integrability, we recall two well-known Killing field reductions of Einstein’s equations, unify them using a harmonic map formulation, and state two results on the integrability of the equations and solvability of the dressing system. The resulting algorithm is then combined with an asymptotic analysis to produce constraints on the degrees of freedom arising in the solution-generation mechanism. The approach is carried out explicitly for the Einstein vacuum equations. Applications of the technique to other geometric field theories are also discussed.