Energy thresholds for the existence of breather solutions and traveling waves on lattices

We discuss the existence of breathers and of energy thresholds for their formation in DNLS lattices with linear and nonlinear impurities. In the case of linear impurities we present some new results concerning important differences between the attractive and repulsive impurity which is interplaying with a power nonlinearity. These differences concern the coexistence or the existence of staggered and unstaggered breather profile patterns. We also distinguish between the excitation threshold (the positive minimum of the power observed when the dimension of the lattice is greater or equal to some critical value) and explicit analytical lower bounds on the power (predicting the smallest value of the power a discrete breather one-parameter family), which are valid for any dimension. Extended numerical studies in one, two and three dimensional lattices justify that the theoretical bounds can be considered as thresholds for the existence of the frequency parametrized families. The discussion reviews and extends the issue of the excitation threshold in lattices with nonlinear impu- rities while lower bounds, with respect to the kinetic energy, are also discussed for traveling waves in FPU periodic lattices

Floquet analysis of Kuznetsov-Ma breathers: A path towards spectral stability of rogue waves

In the present work, we aim at taking a step towards the spectral stability analysis of Peregrine solitons, i.e., wave structures that are used to emulate extreme wave events. Given the space-time localized nature of Peregrine solitons, this is a priori a nontrivial task. Our main tool in this effort will be the study of the spectral stability of the periodic generalization of the Peregrine soliton in the evolution variable, namely the Kuznetsov-Ma breather. Given the periodic structure of the latter, we compute the corresponding Floquet multipliers, and examine them in the limit where the period of the orbit tends to infinity. This way, we extrapolate towards the stability of the limiting structure, namely the Peregrine soliton. We find that multiple unstable modes of the background are enhanced, yet no additional unstable eigenmodes arise as the Peregrine limit is approached. We explore the instability evolution also in direct numerical simulations.Unión Europea MAT2016-79866-R (AEI / FEDER, UE

Discrete solitons in optical BEC lattices. Effects of n-body interactions

In this poster we show some recent results concerning discrete solitons in strong optical lattices, which can be described by the Discrete Nonlinear Schrödinger equation. These results are related to a variation of this equation including saturable nonlinearity terms, a feature throughoutly studied in nonlinear optics. After presenting the derivation of the DNLS equation from the Gross-Pitaevskii equation in the presence of a strong optical lattice, we study the existence of thresholds in the quadratic norm of discrete solitons in the cubic DNLS, cubic-quintic DNLS and photorefractive-DNLS. The second part of the poster is devoted to moving discrete solitons in the photorefractive DNLS equation. In the one hand, we study the existence of radiationless moving discrete solitons; on the other hand, we study the collisions of moving discrete solitons

Escape dynamics in the discrete repulsive φ4 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. © 2012 Elsevier B.V. All rights reserved

Supersonic Kinks in Coulomb lattices

There exist in nature examples of lattices of elements for which the interaction is repulsive, the elements are kept in place because different reasons, as border conditions, geometry (e.g., circular) and, certainly, the interaction with other elements in the system, which provides an external potential. A primer example are layered silicates as mica muscovite, where the potassium ions form a two dimensional lattice between silicate layers. We propose an extremely simplified model of this layer in order to isolate the properties of a repulsive lattice and study them. We find that they are extremely well suited for the propagation of supersonic kinks and multikinks. Theoretically, they may have as much energy and travel as fast as desired. This striking results suggest that the properties of repulsive lattices may be related with some yet not fully explained direct and indirect observations of lattice excitations in muscovite

Kuznetsov–Ma breather-like solutions in the Salerno model

The Salerno model is a discrete variant of the celebrated nonlinear Schr¨odinger (NLS) equation interpolating between the discrete NLS (DNLS) equation and completely integrable Ablowitz-Ladik (AL) model by appropriately tuning the relevant homotopy parameter. Although the AL model possesses an explicit time-periodic solution known as the Kuznetsov-Ma (KM) breather, the existence of time-periodic solutions away from the integrable limit has not been studied as of yet. It is thus the purpose of this work to shed light on the existence and stability of time-periodic solutions of the Salerno model. In particular, we vary the homotopy parameter of the model by employing a pseudo-arclength continuation algorithm where time-periodic solutions are identified via fixed-point iterations. We show that the solutions transform into time-periodic patterns featuring small, yet non-decaying far-field oscillations. Remarkably, our numerical results support the existence of previously unknown time-periodic solutions even at the integrable case whose stability is explored by using Floquet theory. A continuation of these patterns towards the DNLS limit is also discussed.AEI/FEDER (UE) MAT2016-79866-RRegional Government of Andalusia (Spain) P18-RT-348