Discrete Burridge-Knopoff model, with exact solitonic or compactlike traveling wave solution.

We have explored the dynamics of two versions of a Burridge-Knopoff model: with linear or nonlinear interactions between adjacent blocks. We have shown that by properly choosing the analytical form of the discrete solitary wave solution of the model we can calculate analytically the form of the friction function. In both cases our analytical results show that the friction force naturally presents the behavior of a simple weakening friction law first introduced qualitatively by Burridge and Knopoff [Bull. Seismol. Soc. Am. 57, 3411 (1967)] and quantitatively by Carlson and Langer [Phys. Rev. Lett. 62, 2632 (1989)]. With such a force function the discrete solitonic or compactlike wave-front solutions are exact and stable solutions. In the case of linear coupling our numerical simulations show that an irregular initial state evolves into kink pairs (large-amplitude events), that can recombine or not, plus nonlinear localized modes and small linear oscillations (small-amplitude events) that disperse with time, owing to dispersion. For nonlinear coupling one observes compactlike kink pairs or shocks, and a background of robust incoherent nonlinear oscillations (small amplitude events) that persist with time. Our results show that discreteness is a necessary ingredient to observe a rich and complex dynamical behavior. Nonlinearity allows the existence of strictly localized shocks.

Oscillations of a highly discrete breather with a critical regime

We analyze carefully the essential features of the dynamics of a stationary discrete breather in the ultimate degree of energy localization in a nonlinear Klein-Gordon lattice with an on-site double-well potential. We demonstrate the existence of three different regimes of oscillatory motion in the breather dynamics, which are closely related to the motion of the central particle in an effective potential having two nondegenerate wells. In given parameter regions, we observe an untrapped regime, in which the central particle executes large-amplitude oscillations from one to the other side of the potential barrier. In other parameter regions, we find the trapped regime, in which the central particle oscillates in one of the two wells of the effective potential. Between these two regimes we find a critical regime in which the central particle undergoes several temporary trappings within an untrapped regime. Importantly, our study reveals that in the presence of purely anharmonic coupling forces, the breather compactifies, i.e., the energy becomes abruptly localized within the breather.

Breather compactons in nonlinear Klein-Gordon systems.

We demonstrate the existence of a localized breathing mode with a compact support, i.e., a stationary breather compacton, in a nonlinear Klein-Gordon system. This breather compacton results from a delicate balance between the harmonicity of the substrate potential and the total nonlinearity induced by the substrate potential and the coupling forces between adjacent lattice sites.

Dissipative lattice model with exact traveling discrete kink-soliton solutions: discrete breather generation and reaction diffusion regime.

We introduce a nonlinear Klein-Gordon lattice model with specific double-well on-site potential, additional constant external force and dissipation terms, which admits exact discrete kink or traveling wave fronts solutions. In the non-dissipative or conservative regime, our numerical simulations show that narrow kinks can propagate freely, and reveal that static or moving discrete breathers, with a finite but long lifetime, can emerge from kink-antikink collisions. In the general dissipative regime, the lifetime of these breathers depends on the importance of the dissipative effects. In the overdamped or diffusive regime, the general equation of motion reduces to a discrete reaction diffusion equation; our simulations show that, for a given potential shape, discrete wave fronts can travel without experiencing any propagation failure but their collisions are inelastic.

Motion of compactonlike kinks.

We analyze the ability of a compactonlike kink (i.e., kink with compact support) to execute a stable ballistic propagation in a discrete Klein-Gordon system with anharmonic coupling. We demonstrate that the effects of lattice discreteness, and the presence of a linear coupling between lattice sites, are detrimental to a stable ballistic propagation of the compacton, because of the particular structure of the small-oscillation frequency spectrum of the compacton in which the lower-frequency internal modes enter in direct resonance with phonon modes. Our study reveals the parameter regions for obtaining a stable ballistic propagation of a compactonlike kink. Finally we investigate the interactions between compactonlike kinks.