Discrete Klein-Gordon Models Research Articles

Analysis of the response of damped and parametrically driven, strongly anharmonic Klein-Gordon chain - Part 2: Beat-waves

This paper continues a series of publications devoted to the analysis of strongly nonlinear wave phenomena, emerging in genuinely anharmonic, Klein-Gordon cyclic chains, subjected to a traveling-wave parametric excitation. In this study, we analyze a special type of nonlinear waves, which will be further referred to as beat-waves. These are spatially extended, and strongly modulated waves manifested by the extreme amplitude modulation, which can be of stationary or nonstationary type. Stationary beat-waves of Klein-Gordon chains propagate with constant group velocity, that depends on the beat-wave amplitude. In the vicinity of 2:1 parametric resonance, we derive the parametrically driven, discrete p-Schrödinger (PD-DpS) model, which depicts the slow modulation of the response of a parametric oscillator chain. Special coordinates' transformation applied on the PD-DpS model allows the exact reduction of extremely modulated beat-waves to an invariant manifold of this slow-flow system. The system dynamics reduced to the beat-wave invariant manifold is depicted by three collective coordinates. In the undamped case, the analysis of the slow-flow system dynamics on the invariant manifold reveals a special family of nonlinear beat-waves characterized by an extreme amplitude modulation. Among these beat-wave solutions lying on the manifold, our analysis identifies some special solutions manifested by the solitary-pulse like, temporal evolution of the beat-wave amplitude which can either decay to zero or to a constant nonzero value. We derive the exact analytical solutions for these special types of waves as well as obtain an explicit analytical approximation for the beat-waves of other types. In the damped case, analysis of the slow-flow system dynamics on the invariant manifold fully recovers the mechanism of system synchronization on the special, stationary beat-wave solutions characterized by the harmonically modulated envelope. Importantly, the derived reduced order model accurately captures the transient evolution stage as well as the steady-state stage of the collective beat-wave response of PD-DpS model, initialized on the beat-wave invariant manifold. A detailed bifurcation analysis of the stationary undamped as well as the stationary damped beat-waves recovers their entire bifurcation structure and enables to fully characterize the zones of system parameters guaranteeing the existence of this special type of non-linear wave solutions. The results of analytical study of various types of beat wave solutions are in excellent agreement with the results of numerical simulation of both PD-DpS as well as the original, parametrically driven, genuinely an-harmonic, discrete Klein-Gordon model.

Stationary multi-kinks in the discrete sine-Gordon equation

We consider the existence and spectral stability of static multi-kink structures in the discrete sine-Gordon equation, as a representative example of the family of discrete Klein–Gordon models. The multi-kinks are constructed using Lin’s method from an alternating sequence of well-separated kink and antikink solutions. We then locate the point spectrum associated with these multi-kink solutions by reducing the spectral problem to a matrix equation. For an m-structure multi-kink, there will be m eigenvalues in the point spectrum near each eigenvalue of the primary kink, and, as long as the spectrum of the primary kink is imaginary, the spectrum of the multi-kink will be as well. We obtain analytic expressions for the eigenvalues of a multi-kink in terms of the eigenvalues and corresponding eigenfunctions of the primary kink, and these are in very good agreement with numerical results. We also perform numerical time-stepping experiments on perturbations of multi-kinks, and the outcomes of these simulations are interpreted using the spectral results.

Escape dynamics in the discrete repulsive [formula omitted] model

We study deterministic escape dynamics of the discrete Klein–Gordon model with a repulsive quartic on-site potential. Using a combination of analytical techniques, based on differential and algebraic inequalities and selected numerical illustrations, we first derive conditions for collapse of an initially excited single-site unit, for both the Hamiltonian and the linearly damped versions of the system and showcase different potential fates of the single-site excitation, such as the possibility to be “pulled back” from outside the well or to “drive over” the barrier some of its neighbors. Next, we study the evolution of a uniform (small) segment of the chain and, in turn, consider the conditions that support its escape and collapse of the chain. Finally, our path from one to the few and finally to the many excited sites is completed by a modulational stability analysis and the exploration of its connection to the escape process for plane wave initial data. This reveals the existence of three distinct regimes, namely modulational stability, modulational instability without escape and, finally, modulational instability accompanied by escape. These are corroborated by direct numerical simulations. In each of the above cases, the variations of the relevant model parameters enable a consideration of the interplay of discreteness and nonlinearity within the observed phenomenology.

Classification of integrable discrete Klein–Gordon models

The Lie algebraic integrability test is applied to the problem of classification of integrable Klein–Gordon-type equations on quad graphs. The list of equations passing the test is presented, containing several well-known integrable models. A new integrable example is found; its higher symmetry is presented.

Effect of damping on kink ratchets in the Klein–Gordon lattice free of the Peierls–Nabarro potential

Ratchet of kink under harmonic ac driving force in the discrete Klein–Gordon model with asymmetric potential and with a damping term is investigated numerically. A particular feature of the considered discrete model is that supported kinks are free of the Peierls–Nabarro potential (PNp). It is demonstrated that, depending on the strength of damping, the drift velocity of the kink can be positive (weak damping) or negative (strong damping). The absence of PNp results in a weak dependence of kink drift velocity on the discreteness parameter.

Exact static solutions for discreteϕ4models free of the Peierls-Nabarro barrier: Discretized first-integral approach

We propose a generalization of the discrete Klein-Gordon models free of the Peierls-Nabarro barrier derived in Spreight [Nonlinearity 12, 1373 (1999)] and Barashenkov [Phys. Rev. E 72, 035602(R) (2005)], such that they support not only kinks but a one-parameter set of exact static solutions. These solutions can be obtained iteratively from a two-point nonlinear map whose role is played by the discretized first integral of the static Klein-Gordon field, as suggested by Dmitriev [J. Phys. A 38, 7617 (2005)]. We then discuss some discrete phi4 models free of the Peierls-Nabarro barrier and identify for them the full space of available static solutions, including those derived recently by Cooper [Phys. Rev. E 72, 036605 (2005)] but not limited to them. These findings are also relevant to standing wave solutions of discrete nonlinear Schrödinger models. We also study stability of the obtained solutions. As an interesting aside, we derive the list of solutions to the continuum phi4 equation that fill the entire two-dimensional space of parameters obtained as the continuum limit of the corresponding space of the discrete models.

Discrete Klein–Gordon models with static kinks free of the Peierls–Nabarro potential

For the nonlinear Klein-Gordon type models, we describe a general method of discretization in which the static kink can be placed anywhere with respect to the lattice. These discrete models are therefore free of the {\it static} Peierls-Nabarro potential. Previously reported models of this type are shown to belong to a wider class of models derived by means of the proposed method. A relevant physical consequence of our findings is the existence of a wide class of discrete Klein-Gordon models where slow kinks {\it practically} do not experience the action of the Peierls-Nabarro potential. Such kinks are not trapped by the lattice and they can be accelerated by even weak external fields.

Vector nonlinear Klein-Gordon lattices: General derivation of small amplitude envelope soliton solutions

Group velocity and group velocity dispersion for a wave packet in vectorial discrete Klein-Gordon models are obtained by an expansion, based on perturbation theory, of the linear system giving the dispersion relation and the normal modes. We show how to map this expansion on the multiple scale expansion in the real space and how to find nonlinear Schrödinger small amplitude solutions when a nonlinear on-site potential balances the group velocity dispersion effect.

Modulational instabilities in discrete lattices.

We study analytically and numerically modulational instabilities in discrete nonlinear chains, taking the discrete Klein-Gordon model as an example. We show that discreteness can drastically change the conditions for modulational instability; e.g., at small wave numbers a nonlinear carrier wave is unstable to all possible modulations of its amplitude if the wave amplitude exceeds a certain threshold value. Numerical simulations show the validity of the analytical approach for the initial stage of the time evolution, provided that the harmonics generated by the nonlinear terms are considered. The long-term evolution exhibits chaoticlike states.

Localized breather-like solution in a discrete Klein-Gordon model and application to DNA

Non-propagating localized oscillating solutions are studied in the case of a discrete one-dimensional system with the 2–3 power onsite potential. A multiple-scale asymptotic perturbation expansion is not adequate for describing the large amplitude oscillations we consider. Starting from an exact pulse-like solution in the continuum, we derive an approximate oscillating solution. Our numerical simulations reveal that, whereas the excitations are strongly damped in the strong intersite coupling case, they are long-lived in the discrete cases. An additional modulation phenomenon of the oscillation, seen in the simulation, is explained by combination of the breather oscillations with the zero group velocity phonons modes. Then, applying the Green's functions formalism in analogy with a previous work (S. Takeno et al., Prog. Theor. Phys. Suppl. 94 (1988)), we derive a very accurate expression for the discrete breather with a quasi-infinite lifetime. Lastly, we demonstrate how such excitations can explain why the breathers make the dominant contributions in the dynamics of a simple model of DNA.