Discrete sine-Gordon Equation Research Articles

Dynamics of an underdamped Josephson-junction ladder.

We show analytically that the dynamical equations for an underdamped ladder of coupled small Josephson junctions can be approximately reduced to the discrete sine-Gordon equation. As numerical confirmation, we solve the coupled Josephson equations for such a ladder in a magnetic field. We obtain discrete-sine-Gordon-like IV characteristics, including a flux flow and a ``whirling'' regime at low and high currents, and voltage steps which represent a lock-in between the vortex motion and linear ``phasons'', and which are quantitatively predicted by a simple formula. At sufficiently high anisotropy, the fluxons on the steps propagate ballistically.

Soliton pinning by long-range order in aperiodic systems.

We investigate propagation of a kink soliton along inhomogeneous chains with two different constituents, arranged either periodically, aperiodically, or randomly. For the discrete sine-Gordon equation and the Fibonacci and Thue-Morse chains taken as examples, we have found that the phenomenology of aperiodic systems is very peculiar: On the one hand, they exhibit soliton pinning as in the random chain, although the depinning forces are clearly smaller. In addition, solitons are seen to propagate differently in the aperiodic chains than on periodic chains with large unit cells, given by approximations to the full aperiodic sequence. We show that most of these phenomena can be understood by means of simple collective coordinate arguments, with the exception of long range order effects. In the conclusion we comment on the interesting implications that our work could bring about in the field of solitons in molecular (e.g., DNA) chains.

Smoothing of rough surfaces.

Simulations of surface smoothing (healing) by Langevin dynamics in large systems are reported. The surface model is described by a two-dimensional discrete sine-Gordon (solid-on-solid) equation. We study how initially circular terraces decay in time for both zero and finite temperatures and we compare the results of our simulations with various analytical predictions. We then apply this knowledge to the smoothing of a rough surface obtained by heating an initially flat surface above the roughening temperature and then quenching it. We identify three regimes in terms of their time evolution, which we are able to associate with the resulting terrace morphology. The regimes consist of a short initial stage, during which small scale fluctuations disappear; an intermediate, longer time interval, when evolution can be understood in terms of terraces and their interaction; and a final situation in which almost all terraces have been suppressed. We discuss the implications of our results for modeling rough surfaces.

Duality of the Existence of Localized Modes (Low- and High-Frequency Modes) in Pure Nonlinear Lattices –Sine Lattices, Discrete Sine-Gordon Lattices, the Ablowitz-Ladik Lattices and One-Dimensional Lattices with Quartic Anharmonicity–

Dual nature of the existence of localized modes in pure nonlinear lattices is shown as the intrinsic properties of the lattices that cannot be shared with their continuum counterpart. This is a coexistence, for given nonlinearity, of low-and high-frequency localized modes appearing above and below, respectively, of the frequency band of the corresponding linear lattices. As a typical example, this is illustrated for breatherlike mode solutions to sine-lattice equations and discrete sine-Gordon equations using approximate analytical solutions. The duality concept is put into firmer basis by showing the coexistence of such two types of localized modes for exact envelope soliton solutions to the Ablowitz-Ladik equations and for approximate analytical solutions with numerical testing for moving nonlinear localized modes in one-dimensional lattices with hard quartic anharmonicity.

Variational approximations to breather modes in the discrete sine-Gordon equation

Long-lived breather-like modes have been observed in the numerical simulations of discrete sine-Gordon systems. However, little analytical work has been carried out on these systems. We show how to use the Variational form of nonlinear Klein-Gordon equations to find corrections to breather modes caused by discreteness.

Doubly discrete Lagrangian systems related to the Hirota and sine-Gordon equation

We extend the action for evolution equations of KdV and mKdV type which was derived by Capel and Nijhoff to the case of not periodic, but only equivariant phase space variables, introduced by Faddeev and Volkov. The difference of these variables may be interpreted as reduced phase space variables via a Marsden-Weinstein reduction where the monodromies play the role of the momentum map. As an example we obtain the doubly discrete sine-Gordon equation and the Hirota equation and the corresponding symplectic structures.

Whirling modes and parametric instabilities in the discrete Sine-Gordon equation: Experimental tests in Josephson rings.

We analyze the damped driven discrete sine-Gordon equation. For underdamped, highly discrete systems, we show that whirling periodic solutions undergo parametric instabilities at certain drive strengths. The theory predicts novel resonant steps in the current-voltage characteristics of discrete Josephson rings, occurring in the return path of the subgap region. We have observed these steps experimentally in a ring of 8 underdamped junctions. An unusual prediction, verified experimentally, is that such steps occur even if there are no vortices in the ring. Numerical simulations indicate that complex spatiotemporal behavior occurs past the onset of instability.

The discrete quantum pendulum

We study a doubly discrete quantum sine-Gordon equation, which is derived from the quantum Volterra model solved by Faddeev and Volkov [Teor. Mat. Fiz. 92 (1992) 207]. The discrete quantum pendulum is obtained as the simplest special case. The eigenfunctions of its Hamiltonian, displayed as densities on the phase space, are very localized around the classical energy levels.

Chaos and the approach to equilibrium in a discrete sine-Gordon equation

In this paper we study the long time dynamics of a Hamiltonian system with many degrees of freedom. We numerically investigate the system's approach to equipartation of energy when the intial energy is confined to one or a small set of Fourier modes. We find that there is a transition from a low energy regime in which energy does not spread appreciably among the modes to a high energy regime in which the system rapidly approaches equipartition, and that this transition coincides with the onset of chaos in the system, as evidenced by a sharp rise in the largest Lyapunov exponent. For low frequency initial conditions, the critical parameter for this transition is the scaled energy E 1 = ( L/ N) 2 E. Using a generalization of the traditional Chirikov resonance overlap calculation on a three-mode subset of the full system, we predict the onset of widespread chaos and the transition to equipartition on relatively short timescales.

Langevin molecular dynamics of interfaces: Nucleation versus spiral growth.

A simulation of surface growth is reported that directly introduces dynamics and thermal noise within a classical Langevin molecular-dynamics scheme. Using recent advances in massively parallel computation, extensive simulations on large two-dimensional lattices are possible. Surface growth is modeled by a dynamic solid-on-solid model, analogous to a discrete two-dimensional sine-Gordon equation. The influence of both homogeneous (thermal) nucleation and Frank-Read sources of spiral growth patterns are incorporated and compared. A phase diagram is described in the space of temperature and chemical potential difference between surface and vapor. At sufficiently high temperatures, a surface-roughening transition occurs. Finally, the same model is applied to other two-dimensional contexts: charge-density-wave materials in an electric field, and a two-dimensional Josephson junction in a perpendicular current.

A unified algebraic approach to integral and discrete evolution equations

The authors investigate the integrability properties of discrete systems and singular integral evolution equations, showing that they correspond to different reductions and different concrete realisations of the same abstract algebraic structures. In particular, they derive the bi-Hamiltonian formulation of prototype examples like the discrete sine-Gordon equation and the sine-Hilbert equation.

Parametric instabilities in the discrete sine-Gordon equation

The one-dimensional sine-Gordon equation is an example of an exactly solvable nonlinear partial differential equation. When discretized on a periodic lattice in space and in time, it corresponds to a lattice of pendula coupled by linear springs. We show that the discretized system is unstable to a parametric mode when the frequency of time discretization is sufficiently small, and we obtain the instability condition and growth rate. By considering the effects of two finite amplitude modes, we also obtain the nonlinear saturation of the instability. We also examine how solutions of the exact sine-Gordon equation behave under this map, both in and away from the parametrically unstable regime.

Discrete bäcklund transformation

A discretized Bäcklund transformation is given for the discrete sine-Gordon equation, and the general soliton solution is obtained.

A Sine-Lattice (Sine-Form Discrete Sine-Gordon) Equation–One-and Two-Kink Solutions and Physical Models–

A sine-lattice (SL) (sine-form discrete sine-Gordon) equation defined as \(\sin (u_{n+1}-u_{n})-\sin (u_{n}-u_{n-1})-\ddot{u}_{n}{=}g \sin u_{n}\) is studied. By introducing a dependent variable transformation and the Hirota D–operators, it can be recast into a bilinear operator form similar to that of the sine-Gordon (SG) equation. This shows that the SL equation is much closer in its soliton properties to the SG equation than the discrete SG equation with the difference factor u n +1 + u n -1 -2 u n . It is shown that the SL equation yields, though not mathematically exact, but well-defined, one-kink and specific two-kink solutions. A numerical calculation which lends support to this is also presented. Several physical models are discussed where the SL equation appears as a more natural model field equation than the discrete SG equation.

Numerical Evidence of the Existence of Approximate Kink-Antikink States and Breather-Like Modes in Sine-Form Discrete Double Sine-Gordon Equation

On effectue une etude numerique de la collision frontale solution duale ― solution antiduale pour l'equation de sinus-Gordon discrete double

The inverse scattering transform for solving the discrete Sine-Gordon equation

The inverse scattering transform for solving the discrete Sine-Gordon equation

Discontinuous Solitons for Discrete Sine-Gordon Equation

It is demonstrated that the semi-discrete Sine-Gordon equation exhibits discontinuous soliton solutions. Three types of new solutions are presented and subjected to examination.

Pinning transition of the discrete sine-Gordon equation

The ground state of the discrete sine-Gordon equation, used to model a one-dimensional solid in a periodic potential, is examined in the incommensurate region. The behavior of the system near the transition from an unpinned to a pinned phase (first discussed by Aubry) is investigated. A disorder parameter and a correlation length are defined and shown numerically to obey scaling relations on both sides of the transition. The system studied is equivalent to the "standard map" of dynamical systems theory, and this relationship is discussed. In particular, our results extend the scaling behavior found by Shenker and Kadanoff into the "chaotic" regime.

The inverse scattering transform for solving the discrete sine-Gordon equation

In this letter we develop, in a concise way, the generalization of the discrete Zakharov and Shabat spectral transform necessary to solve the discrete sine-Gordon equation and, at the end, recover its soliton solutions.

Group-theoretical aspects of the discrete sine-Gordon equation

The group-theoretical interpretation of the sine-Gordon equation in terms of connection forms on fiber bundles is extended to the discrete case. Solutions of the discrete sine-Gordon equation induce surfaces on a lattice in the SU(2) group space. The inverse scattering representation, expressing the parallel transport of fibers, is implemented by means of finite rotations. Discrete B\"acklund transformations are realized as gauge transformations. The three-dimensional inverse scattering representation is used to derive a discrete nonlinear $\ensuremath{\sigma}$ model, and the corresponding B\"acklund transformation and Pohlmeyer's $R$ transformation are constructed.