In this work the interaction between a spatial localized and time periodic state (discrete breather) and small amplitude plane waves is studied.

In the mid-1920s, the great Albert Einstein proposed that at extremely low temperatures, a gas of bosonic particles will enter a new phase where a large fraction of them occupy the same quantum state. This state would bring many of the peculiar features of quantum mechanics, previously reserved for small samples consisting only of a few atoms or molecules, up to a macroscopic scale. This is what we today call a Bose-Einstein condensate. It would take physicists almost 70 years to realize Einstein's idea, but in 1995 this was finally achieved. The research on Bose-Einstein condensates has since taken many directions, one of the most exciting being to study their behavior when they are placed in optical lattices generated by laser beams. This has already produced a number of fascinating results, but it has also proven to be an ideal test-ground for predictions from certain nonlinear lattice models. Because on the other hand, nonlinear science, the study of generic nonlinear phenomena, has in the last half century grown out to a research field in its own right, influencing almost all areas of science and physics. Nonlinear localization is one of these phenomena, where localized structures, such as solitons and discrete breathers, can appear even in translationally invariant systems. Another one is the (in)famous chaos, where deterministic systems can be so sensitive to perturbations that they in practice become completely unpredictable. Related to this is the study of different types of instabilities; what their behavior are and how they arise. In this thesis we compare classical and quantum mechanical nonlinear lattice models which can be applied to BECs in optical lattices, and also examine how classical nonlinear concepts, such as localization, chaos and instabilities, can be transfered to the quantum world.

In this work the interaction between a spatial localized and time periodic state (discrete breather) and small amplitude plane waves is studied.

<p>We consider existence and stability of breather solutions to discrete nonlinear Schrodinger (dNLS) and discrete Klein-Gordon (dKG) equations near the anti-continuum limit, the limit of the zero coupling constant. For sufficiently small coupling, discrete breathers can be uniquely extended from the anti-continuum limit where they consist of periodic oscillations on excited sites separated by "holes" (sites at rest).</p> <p>In the anti-continuum limit, the dNLS equation linearized about its discrete breather has a spectrum consisting of the zero eigenvalue of finite multiplicity and purely imaginary eigenvalues of infinite multiplicities. Splitting of the zero eigenvalue into stable and unstable eigenvalues near the anti-continuum limit was examined in the literature earlier. The eigenvalues of infinite multiplicity split into bands of continuous spectrum, which, as observed in numerical experiments, may in turn produce internal modes, additional eigenvalues on the imaginary axis. Using resolvent analysis and perturbation methods, we prove that no internal modes bifurcate from the continuous spectrum of the dNLS equation with small coupling.</p> <p>Linear stability of small-amplitude discrete breathers in the weakly-coupled KG lattice was considered in a number of papers. Most of these papers, however, do not consider stability of discrete breathers which have "holes" in the anti-continuum limit. We use perturbation methods for Floquet multipliers and analysis of tail-to-tail interactions between excited sites to develop a general criterion on linear stability of multi-site breathers in the KG lattice near the anti-continuum limit. Our criterion is not restricted to small-amplitude oscillations and it allows discrete breathers to have "holes" in the anti-continuum limit.</p> / Doctor of Philosophy (PhD)

Μελετάµε χωρικά εντοπισµένες και χρονικά περιοδικές λύσεις σε διακριτά συστήµατα που εκτείνονται σε µία χωρική διάσταση. Αυτού του είδους οι λύσεις είναι γνωστές µε τον όρο discrete breathers (DB) ή intrinsic localized modes (ILM). Στην ελληνική ϐιϐλιογραϕία, έχουν ονοµαστεί ∆ιακριτές Πνοές. Απαραίτητα χαρακτηριστικά για την εµϕάνιση τέτοιων λύσεων είναι η ύπαρξη ενός άνω φράγµατος του γραµµικού φάσµατος καθώς και η µη γραµµικότητα των εξισώσεων κίνησης, χαρακτηριστικά που συναντάµε σε πολλά φυσικά συστήµατα. Συγκεκριμένα, ασχολούµαστε µε πλέγµατα τύπου Klein Gordon και παρουσιάσουµε μια αποδείξη ύπαρξης τέτοιων λύσεων καθώς και αριθµητικά αποτελέσµατα µελετώντας παράλληλα την ευστάθεια των περιοδικών αυτών λύσεων µέσω της ϑεωρίας Floquet. Πέραν του κλασικού µοντέλου, όπου έχουµε αλληλεπιδράσεις πλησιέστερων γειτόνων, εισάγουµε επίσης ένα νέο µοντέλο µε αλληλεπιδράσεις µακράς εµβέλειας η οποία ελέγχεται µέσω µιας παράµετρου α και µελετάµε τις επιπτώσεις που έχει η μεταβολή του εύρους αλληλεπίδρασης στον χωρικό εντοπισµό και την ευστάθεια ενός DB. / We study time-periodic and spatially localized solutions in discrete dynamical systems describing Hamiltonian lattices in one spatial dimension. These solutions are called discrete breathers (DBs) or intrinsic localized modes (ILM). Necessary conditions for their occurrence are the boundedness of the spectrum of linear oscillations of the system as well as the nonlinearity of the equations of motion. More specifically, we focus on a Klein Gordon lattice and present an existence proof for such solutions, as well as numerical results revealing the stability (or instability) of DBs using Floquet theory. Besides reporting on the classical Klein Gordon model with nearest neighbor interactions, we also introduce long range interactions in our model, which are controlled by a parameter α and study the effect of varying the range of interactions on the spatial localization and the stability of a DB.