Escape dynamics in the discrete repulsive model

Abstract

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.

Nonlinear Instabilities of Multi‐Site Breathers in Klein–Gordon Lattices

“…The solution u of the KG lattice equation (1) is a function of time t. A local solution u exists in C 1 ((−T, T ); 2 (Z)) for some T > 0 if V is Lipschitz. This can be easily deduced from the Picard's contraction method, thanks to the boundedness of the discrete Laplace operator in the lattice equation (1).…”

PT-symmetric dimer of coupled nonlinear oscillators

“…This is strongly related to the escape dynamics considered in Ref. [30]. On the contrary, we want to highlight that this is different than the "worst case scenario" of the Schrödinger dimer of the RWA.…”

“…On the contrary, the latter model has the potential for finite-time collapse when the amplitude of the nodes exceeds the unit height of the potential (see a detailed analysis of this "escape" phenomenology in the recent work of Ref. [30] and references therein). It is thus rather natural that the two models should significantly deviate from each other as this parameter range is approached.…”

“…For > 0, this nonlinearity is soft, imposing a finite (maximal energy) height type of potential, enabling the possibility of indefinite growth by means of the escape scenario considered earlier, e.g., in Ref. [30] (see also references therein). On the other hand, for < 0, the nonlinearity is hard, and the potential is monostable, bearing only the ground state at 0 and no possibility for such finite-time collapse (in the Hamiltonian analog of the model, only the potential for oscillations around the 0 state exists in this case of the hard potential).…”

“…On the other hand, in terms of substantial differences, it is naturally expected that for soft nonlinearities, the oscillator model can escape the potential well (and, hence, have collapse features) even when the Schrödinger model cannot; see for a detailed recent discussion of such features in the Hamiltonian limit the work of Ref. [30]. More importantly for our purposes, another significant difference is that it turns out that both branches, namely both the symmetric and the antisymmetric one, have destabilizing symmetry-breaking bifurcations, even though in the Schrödinger reduction, only one of the two branches (the symmetric for soft and the antisymmetric for hard, as mentioned above) sustains such bifurcations.…”