Discrete breathers in Φ4 and related models

In this Chapter, we touch upon the wide topic of discrete breather (DB) formation with a special emphasis on the prototypical system of interest, namely the 4 model. We start by introducing the model and discussing some of the application areas/motivational aspects of exploring time periodic, spatially localized structures, such as the DBs. Our main emphasis is on the existence, and especially on the stability features of such solutions.We explore their spectral stability numerically, as well as in special limits (such as the vicinity of the so-called anti-continuum limit of vanishing coupling) analytically. We also provide and explore a simple, yet powerful stability criterion involving the sign of the derivative of the energy vs. frequency dependence of such solutions. We then turn our attention to nonlinear stability, bringing forth the importance of a topological notion, namely the Krein signature. Furthermore, we briefly touch upon linearly and nonlinearly unstable dynamics of such states. Some special aspects/extensions of such structures are only touched upon, including moving breathers and dissipative variations of the model and some possibilities for future work are highlighted. While this Chapter by no means aspires to be comprehensive, we hope that it provides some recent developments (a large fraction of which is not included in time-honored DB reviews) and associated future possibilities.AEI/FEDER, (UE) MAT2016- 79866-

Discrete breathers collisions: An overview and some recent results

We present some results on breather collisions in DNLS lattices, with special focus on systems with saturable nonlinearity. We also review some other works concerning to collisons in FPU and Klein-Gordon lattices, proposing future challenges

Solitons for the cubic-quintic nonlinear Schrödinger equation with time and space modulated coefficients

In this paper, we construct, by means of similarity transformations, explicit solutions to the cubic–quintic nonlinear Schrödinger equation with potentials and nonlinearities depending on both time and spatial coordinates. We present the general approach and use it to calculate bright and dark soliton solutions for nonlinearities and potentials of physical interest in applications to Bose–Einstein condensates and nonlinear optics.project FIS2008-02873 Ministerio de Ciencia e Innovació

Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics

In this paper, we give a proof of the existence of stationary dark solitonsolutions or heteroclinic orbits of nonlinear equations of Schrödinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of dark solitons for cubic and photorefractive nonlinearities.MICINN project FIS2008-0484

Ratchet effect in solids: Defect transport driven by biharmonic forces

PósterInterstitials and vacancies, in one-dimensional lattices, are point defects that can be modelled by means of kinks or antikinks in a discrete Frenkel-Kontorova model, with a sine-Gordon on-site potential. The properties of kinks and antikinks are the same if a harmonic interaction potential is considered. The ratchet properties of these defects in the above mentioned context has been studied by Zolotaryuk and Salerno when the system is driven by a biharmonic field. The properties of these solutions are strongly altered when an anharmonic interaction potential is introduced in the model, as the Peierls-Nabarro barrier is higher for antikinks than for kinks. The aim of this poster is to show the effects of the anharmoniciy of the interaction potential in the properties of kinks and antikinks focusing in the assymetry between the properties of these two species of topological solitonsMinisterio de Educación y Ciencia FIS2006-27277-

An energy-based stability criterion for solitary traveling waves in Hamiltonian lattices

In this work, we revisit a criterion, originally proposed in [Nonlinearity {\bf 17}, 207 (2004)], for the stability of solitary traveling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-traveling frame, as well as at that of the Floquet problem arising when considering the traveling wave as a periodic orbit modulo a shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the traveling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed

Propagation studies for the construction of atomic macro-coherence in dense media as a tool to investigate neutrino physics

In this manuscript we review the possibility of inducing large coherence in a macroscopic dense target by using adiabatic techniques. For this purpose we investigate the degradation of the laser pulse through propagation, which was also related to the size of the prepared medium. Our results show that, although adiabatic techniques offer the best alternative in terms of stability against experimental parameters, for very dense media it is necessary to engineer laser-matter interaction in order to minimize laser field degradation. This work has been triggered by the proposal of a new technique, namely Radiative Emission of Neutrino Pairs (RENP), capable of investigating neutrino physics through quantum optics concepts which require the preparation of a macrocoherent state.Ministerio de Economía y Competitividad (Spain) FIS2014-53371-C4-3-RMinisterio de Economía y Competitividad (Spain) FIS2014-53371-C4-1-

A PT-Symmetric Dual-Core System With the Sine-Gordon Nonlinearity and Derivative Coupling

We propose a system of sine-Gordon equations, with the symmetry represented by balanced gain and loss in them. The equations are coupled by sine-field terms and first-order derivatives. The sinusoidal coupling stems from local interaction between adjacent particles in the coupled Frenkel-Kontorova (FK) chains and related sine-lattices, while the cross-derivative coupling, which was not considered before, is induced by \emph{three-particle} interactions, provided that the particles in the parallel FK\ chains move in different directions. Nonlinear wave structures are then studied in this model. In particular, kink-kink (KK) and kink-antikink (KA) complexes are explored by means of analytical and numerical methods. It is predicted analytically and confirmed numerically that the complexes are unstable for one sign of the sinusoidal coupling, and stable for another. Stability regions are delineated in the underlying parameter space. Unstable complexes split into free kinks/antikinks that may propagate or become stationary, depending on whether they are subject to gain or loss, respectively

Nonlinearity and Topology

The interplay of nonlinearity and topology results in many novel and emergent properties across a number of physical systems such as chiral magnets, nematic liquid crystals, Bose-Einstein condensates, photonics, high energy physics, etc. It also results in a wide variety of topological defects such as solitons, vortices, skyrmions, merons, hopfions, monopoles to name just a few. Interaction among and collision of these nontrivial defects itself is a topic of great interest. Curvature and underlying geometry also affect the shape, interaction and behavior of these defects. Such properties can be studied using techniques such as, e.g. the Bogomolnyi decomposition. Some applications of this interplay, e.g. in nonreciprocal photonics as well as topological materials such as Dirac andWeyl semimetals, are also elucidated.AEI/FEDER, (UE) MAT2016-79866-