<p>Ispitivan je generalisani Frenkel-Kontorova model pod&nbsp;dejstvom spolja&scaron;njih periodičnih sila,&nbsp;i njihov uticaj na &Scaron;apiro stepenike.&nbsp;Deformacija substratnog potencijala prouzrokuje&nbsp;pojavu velikih polucelih i drugih subharmonijskih&nbsp;stepenika u sistemu.&nbsp;Rezultati dobijeni u radu su od značaja&nbsp;u talasima i spinovima gustine naboja&nbsp;kao i u Džozefsonovim spojevima.</p> / <p>Properties of the Shapiro steps in dierent<br />commensurate structures of the ac + dc driven<br />Frenkel-Kontorova model with dierent generalized<br />substrate potential have been examined.<br />Deviation from the sinusoidal potential produces<br />appearance of large subharmonic steps.<br />Presented results could be of great importance<br />for thestudies of all real systems closely related<br />to the dissipative dynamics of the FK model<br />such as studies of the charge- or spin-density wave<br />systems and the systems of Josephson-junction arrays.</p>

We discuss the twist map, with a special interest in its use for the finite Frenkel–Kontorova model. We explain the meaning of the tensile force in some proposed models. We demonstrate that the application of the twist map for the finite FK model is not correct, because the procedure ignores the necessary boundary conditions.

This thesis, submitted to the University of Manchester, covers a range of topics related to current research in two-dimensional materials under the title: 'Investigations into the interfacial interaction of graphene with hexagonal boron nitride.'In the last decade, two-dimensional materials have become a rich source of original research and potential applications. The main advantage lies in the ability to produce novel composite structures, so-called 'layered heterostructures', which are only a few atomic layers thick. One can utilise the unique properties of several species of crystal separately, or how they interact to realise a diverse range of uses. Two such crystals are graphene and hexagonal boron nitride. Hexagonal boron nitride has, so far, been used primarily as a substrate for graphene, allowing researchers to get the most out of graphene's impressive individual properties. However, in this thesis, the non-trivial van der Waals interaction between graphene and hexagonal boron nitride is examined. The interface potential reveals itself as a relatively large-scale, orientation-dependant superlattice, which is described in chapters 1 and 2.I In Chapter 4, the effect of this superlattice is examined by measurement of its effect upon the electrons in graphene, where its modulation leads to the creation of second and third generation Dirac points, revealing Hofstadter's Butterfly. As well as an excellent example of the physics possible with graphene, it also presents a new tool with which to create novel devices possessing tailored electronic properties. II In chapter 5, the consequential effect of the superlattice potential on the structure of graphene is studied. Results are discussed within the framework of the Frenkel-Kontorova model for a chain of atoms on a static background potential. Results are consistent with relaxation of the graphene structure leading to the formation of a commensurate ground state. This has exciting consequences for the production of heterostructures by demonstrating that alignment angle can have large effects upon the physical properties of the crystals. III In chapter 6, the van der Waals potential is shown to be responsible for the self-alignment of the two crystals. This effect is important for the fabrication of perfectly aligned devices and may lead to new applications based on nanoscale motion.

Cette thèse porte sur la construction de solutions particulières de type ondes progressives ou ondes planes pour différentes équations aux dérivées partielles (EDP) et en particulier le modèle de Frenkel-Kontorova et une équation des ondes régularisée. Ce mémoire s’articule comme suit. Le chapitre 1 est destiné à une introduction générale dans laquelle je présente une motivation physique et un résumé de mon travail. Le chapitre 2 est destiné à l’étude d’existence et d’unicité des ondes progressives avec le terme d’accélération. Ce modèle consiste d’un système d’ODE qui décrit le mouvement de particules en interaction. Les applications les plus importantes que nous avons à l’esprit est le mouvement des défauts cristallins appelés dislocations. Pour ce modèle, nous montrons l’existence des ondes progressives sous des hypothèses très faibles. L’unicité de la vitesse a été étudiée ainsi que l’unicité du profil en utilisant les différents types du principe de maximum fort. Comme ce que nous savons, c’est le premier résultat concernant les ondes progressives pour un système accéléré, spatialement discret. Ce chapitre est un article publié à la revue Journal of Dynamic and Differential Equation : Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model, Journal of Dynamic and Differential Equation : Volume 26, Issue 24 (2014), page 1133-1169. Le chapitre 3 est réservé à l’homogénéisation numérique du modèle Frenkel-Kontrova dans le cas amortie. Je présente deux méthodes pour calculer l’hamiltonien effectif: la méthode grand temps et la méthode de Newton. Quelques simulations de l’hamiltonien effectif sont fournies. Le chapitre 4 est destiné à l’étude d’équation d’onde dans un domaine périodique. Selon certaines hypothèses, je construis une solution d’onde plane pour le problème approché et je montre que cette solution satisfait certaines propriétés. Je définis un opérateur non local et un terme correcteur afin de contrôler les oscillations de la solution dans l’espace et dans le temps. Je prouve la construction d’une solution d’onde plane pour un problème approché en utilisant la notion de solution de viscosité. / This thesis deals with the construction of particular solutions of traveling wave or plane wave for different equations partial derivative (EDP) and in particular the Frenkel-Kontorova model and a regularized wave equation. This memory is structured as follows. The chapter 1 is preserved for a general introduction in which i present a physical motivation and a abstract of my work. In chapter 2, I interested to the study the existence and uniqueness of traveling wave solution for the accelerated Frenkel-Kontorova model. This model consist in a system of ODE that describe the motion particles in interaction. The most important applications ihave inmind in the motion of cristal defects called dislocations. For this model, i prove the exxistence of traveling wave solutions under very weak assumptions. The uniqueness of the velocity is also studied as well the uniqueness of the profile which used ddifferent types of strpng maximum principle. As far as we know, this is the first result concerning traveling waves for accelerated, spatially discrete system. This chapter is an article published in the Journal Dynamic and Differential Equation:Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model, Journal of dynamic and Differential Equation : Volume 26, Issue 24 (2014), page 1133-1169. In chapter 3, i interested in the numerical homogenization of fully overdomped frenkel-Kontorova model. I present two methods for computing the effective hamiltonian : large time method and Newton-like method. Some simulations of the effective hamiltonian are provided. Le chapter 4 is preserved to the study a wave equation in a periodic medium. Under certain assumption, i construct a plane wave like solution, and show that this solution satisfy some properties. I define a non- local operator and a term corrector in order to control the oscillations of the solution in space and in time. We prove the construction of a plane wave like solution for the approched problem using the notion of viscosity solution.