It is well known that ultrashort, high intensity pulses with peak powers exceedinga certain critical value (Pcr) undergo self-focusingleading to collapse and filamentation. During the initial stagesof propagation at low intensities the beamdynamics are dominated by diffraction and dispersion. During filamentation, self-focusing resulting from the nonlinear Kerr effect is balanced by higher order nonlinearities such as plasma induced defocusing and absorption.This work examines the role that linear processes combined with initial spatial and temporal conditioningplay in the generation and subsequent evolution of filaments within nonlinearbeams. It is demonstrated that, because of linear diffraction, initial spatial beam shaping can have a dramatic effect on the filament pattern, the number of filaments and the energy contained in each filament. These ideas are applicable to cases that arequite common, such as circularly apodized beams, and help to explain interestingbehavior observed in these types of beams. Finally, it is demonstrated thatwith appropriate preconditioning of multiple subcritical pulses, linear effects can be employed to accurately control when and where filamentation occurs during long distance propagation through conditional collapse of overlapping pulses.

In this dissertation we present theoretical and experimental investigation of the performance quality of fiber optical communication systems, and find new and inexpansive ways of increasing the rate of theinformation transmission.The first part of this work discuss the two major factors limiting the quality of information channels in the fiber optical communication systems. Using methods of large deviation theory from statisticalphysics, we carry out analytical and numerical study of error statistics in optical communication systems in the presence of the temporal noise from optical amplifiers and the structural disorder of optical fibers. In the slowly varying envelope approximation light propagation through optical fiber is described by Schr\{o}dinger's equation. Signal transmission is impeded by the additive (amplifiers) and multiplicative (birefringence) noise This results in signal distortion that may lead to erroneous interpretation of the signal. System performance is characterized by the probability of error occurrence. Fluctuation of spacial disorder due to changing external factors (temperature, vibrations, etc) leads to fluctuations of error rates. Commonly the distribution of error rates is assumed to be Gaussian. Using the optimal fluctuation method we show that this distribution is in fact lognormal. Sucha distribution has ""fat"" tails implying that the likelihood of system outages is much higher than itwould be in the Gaussian approximation. We present experimental results that provide excellent confirmation of our theoretical predictions.In the second part of this dissertation we present some published work on bisolitons in the dispersion managed systems. Modern communication systems use light pulses to transmit tremendous amounts of information. These systems can be modeled using variations of the Nonlinear Shrodinger Equation where chromatic dispersion and nonlinear effects in the glass fiber are taken into account. The best system performance to date is achieved using dispersion management. We will see how the dispersion management works and how it can be modeled. As you pack information more tightly the interaction between the pulsesbecomes increasingly important. In Fall 2005, experiments in Germany showed that bound pairs of pulses (bisolitons) could propagate significant distances. Through numerical investigation we found parametric bifurcation of bisolitonic solutions, and developed a new iterative method with polynomial correction for the calculation of these solutions. Using these solutions in the signal transmission could increase the transmission rates.

This thesis is concerned with analysis of spectral and orbital stability of solitary wave solutions to discrete and continuous PT-symmetric nonlinear Schroedinger equations. The main tools of this analysis are inspired by Hamiltonian systems, where conserved quantities can be used for proving orbital stability and Krein signature can be computed for prediction of instabilities in the spectrum of linearization. The main results are obtained for the chain of coupled pendula represented by a discrete NLS model, and for the trapped atomic gas represented by a continuous NLS model. Analytical results are illustrated with various numerical examples. / Thesis / Doctor of Philosophy (PhD)

The present thesis is devoted to the investigation of certain aspects of the large time behavior of the solutions of two nonlinear Schrödinger equations in dimension three in some suitable perturbative regimes. The first model consist in a Schrödinger equation with a concentrated nonlinearity obtained considering a {point} (or contact) interaction with strength $alpha$, which consists of a singular perturbation of the Laplacian described by a self adjoint operator $H_{alpha}$, and letting the strength $alpha$ depend on the wave function: $ifrac{du}{dt}= H_alpha u$, $alpha=alpha(u)$.It is well-known that the elements of the domain of a point interaction in three dimensions can be written as the sum of a regular function and a function that exhibits a singularity proportional to $|x - x_0|^{-1}$, where $x_0$is the location of the point interaction. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of itssingular part, then, in order to introduce a nonlinearity, we let the strength $alpha$ depend on $u$ according to the law $alpha=-nu|q|^sigma$, with $nu > 0$. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. In particular, we study orbital and asymptotic stability of standing waves for such a model. We prove the existence of standing waves of the form $u (t)=e^{iomega t}Phi_{omega}$, which are orbitally stable in the range $sigma in (0,1)$, and orbitally unstable for $sigma geq 1.$ Moreover, we show that for $sigma in(0,frac{1}{sqrt 2}) cup left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right)$ every standing wave is asymptotically stable, in the following sense. Choosing an initial data close to the stationary state in the energy norm, and belonging to a natural weighted $L^p$ space which allows dispersive stimates, the following resolution holds: $u(t) =e^{iomega_{infty} t +il(t)} Phi_{omega_{infty}}+U_t*psi_{infty} +r_{infty}$, where $U_t$ is the free Schrödinger propagator,$omega_{infty} > 0$ and $psi_{infty}$, $r_{infty} inL^2(R^3)$ with $| r_{infty} |_{L^2} = O(t^{-p}) quadtextrm{as} ;; t right arrow +infty$, $p = frac{5}{4}$,$frac{1}{4}$ depending on $sigma in (0, 1/sqrt{2})$, $sigma in (1/sqrt{2}, 1)$, respectively, and finally $l(t)$ is a logarithmic increasing function that appears when $sigma in (frac{1}{sqrt{2}},sigma^*)$, for a certain $sigma^* in left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right]$. Notice that in the present model the admitted nonlinearities for which asymptotic stability of solitons is proved, are subcritical in the sense that it does not give rise to blow up, regardless of the chosen initial data. The second model is the energy critical focusing nonlinear Schrödinger equation $i frac{du}{dt}=-Delta u-|u|^4 u$. In this case we prove, for any $nu$ and $alpha_0$ sufficiently small, the existence of radial finite energy solutions of the form$u(t,x)=e^{ialpha(t)}lambda^{1/2}(t)W(lambda(t)x)+e^{iDeltat}zeta^*+o_{dot H^1} (1)$ as $tright arrow +infty$, where$alpha(t)=alpha_0ln t$, $lambda(t)=t^{nu}$,$W(x)=(1+frac13|x|^2)^{-1/2}$ is the ground state and $zeta^*$is arbitrarily small in $dot H^1$

Cette thèse est consacrée à l'étude de certains aspects du comportement en temps longs des solutions de deux équations de Schrödinger non-linéaires en dimension trois dans des régimes perturbatives convenables. Le premier modèle consiste en une équation de Schrödinger avec une non-linéarité concentrée obtenue en considérant une interaction ponctuelle de force $alpha$, c'est-à-dire une perturbation singulière du Laplacien décrite par un opérateur autoadjoint $H_{alpha}$, où la force $alpha$ dépend de la fonction d'onde : $ifrac{du}{dt}= H_alpha u$, $alpha=alpha(u)$. Il est bien connu que les éléments du domaine d'une interaction ponctuelle en trois dimensions peuvent être décrits comme la somme d'une fonction régulière et d'une fonction ayant une singularité proportionnelle à $|x - x_0|^{-1}$, où $x_0$ est l'emplacement du point d'interaction. Si $q$ est la charge d'un élément du domaine $u$, c'est-à-dire le coefficient de sa partie singulière, alors pour introduire une non-linéarité, on fait dépendre la force $alpha$ de $u$ selon la loi $alpha=-nu|q|^sigma$, avec $nu > 0$. Ce modèle est défini comme une équation de Schrödinger non-linéaire focalisant de type puissance avec une non-linéarité concentrée en $x_0$. Notre étude regarde la stabilité orbitale et asymptotique des ondes stationnaires de ce modèle. Nous prouvons l'existence d'ondes stationnaires de la forme $u (t)=e^{iomega t}Phi_{omega}$, qui soient orbitalement stables pour $sigma in (0,1)$ et orbitalement instables quand $sigma geq 1.$ De plus nous montrons que si $sigma in (0,frac{1}{sqrt 2}) cup (frac{1}{sqrt 2}, 1)$, alors chaque onde stationnaire est asymptotiquement stable, à savoir que pour des données initiales proches d'un état stationnaire dans la norme d'énergie et appartenant à un espace $L^p$ pondéré où les estimations dispersives sont valides, l'affirmation suivante est vérifiée : il existe $omega_{infty} > 0$ et $psi_{infty} in L^2(R^3)$ tel que $psi_{infty} = O_{L^2}(t^{-p})$ quand $t rightarrow +infty$, tel que $u(t) = e^{iomega_{infty} t +il(t)} Phi_{omega_{infty}} +U_t*psi_{infty} +r_{infty}$, où $U_t$ est le propagateur de Schrödinger libre, $p = frac{5}{4}$, $frac{1}{4}$ respectivement en fonction de $sigma in (0, 1/sqrt{2})$, $sigma in left( frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right)$, et $l(t)$ est une fonction à croissance logarithmique qui apparaît quand $sigma in (frac{1}{sqrt{2}}, sigma^*)$, où $sigma^* in left( frac{1}{sqrt{2}},frac{sqrt{3} +1}{2sqrt{2}} right]$. Notons que dans ce modèle les non-linéarités pour lesquelles on a la stabilité asymptotique sont sous-critiques dans le sens où quelle que soit la donnée initiale il n'y a pas de solutions explosives. Quant au deuxième modèle, il s'agit de l'équation de Schrödinger non-linéaire focalisant à énergie critique : $i frac{du}{dt}=-Delta u-|u|^4 u$. Pour ce cas, nous prouvons, pour tout $nu$ et $alpha_0$ suffisamment petits, l'existence de solutions radiales à énergie finie de la forme $u(t,x)=e^{ialpha(t)}lambda^{1/2}(t)W(lambda(t)x)+e^{iDelta t}zeta^*+o_{dot H^1} (1)$ tout $trightarrow +infty$, où $alpha(t)=alpha_0ln t$, $lambda(t)=t^{nu}$, $W(x)=(1+frac13|x|^2)^{-1/2}$ est l'état stationnaire et $zeta^*$ est arbitrairement petit en $dot H^1$ / The present thesis is devoted to the investigation of certain aspects of the large time behavior of the solutions of two nonlinear Schrödinger equations in dimension three in some suitable perturbative regimes. The first model consist in a Schrödinger equation with a concentrated nonlinearity obtained considering a {point} (or contact) interaction with strength $alpha$, which consists of a singular perturbation of the Laplacian described by a self adjoint operator $H_{alpha}$, and letting the strength $alpha$ depend on the wave function: $ifrac{du}{dt}= H_alpha u$, $alpha=alpha(u)$.It is well-known that the elements of the domain of a point interaction in three dimensions can be written as the sum of a regular function and a function that exhibits a singularity proportional to $|x - x_0|^{-1}$, where $x_0$is the location of the point interaction. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of itssingular part, then, in order to introduce a nonlinearity, we let the strength $alpha$ depend on $u$ according to the law $alpha=-nu|q|^sigma$, with $nu > 0$. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. In particular, we study orbital and asymptotic stability of standing waves for such a model. We prove the existence of standing waves of the form $u (t)=e^{iomega t}Phi_{omega}$, which are orbitally stable in the range $sigma in (0,1)$, and orbitally unstable for $sigma geq 1.$ Moreover, we show that for $sigma in(0,frac{1}{sqrt 2}) cup left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right)$ every standing wave is asymptotically stable, in the following sense. Choosing an initial data close to the stationary state in the energy norm, and belonging to a natural weighted $L^p$ space which allows dispersive stimates, the following resolution holds: $u(t) =e^{iomega_{infty} t +il(t)} Phi_{omega_{infty}}+U_t*psi_{infty} +r_{infty}$, where $U_t$ is the free Schrödinger propagator,$omega_{infty} > 0$ and $psi_{infty}$, $r_{infty} inL^2(R^3)$ with $| r_{infty} |_{L^2} = O(t^{-p}) quadtextrm{as} ;; t right arrow +infty$, $p = frac{5}{4}$,$frac{1}{4}$ depending on $sigma in (0, 1/sqrt{2})$, $sigma in (1/sqrt{2}, 1)$, respectively, and finally $l(t)$ is a logarithmic increasing function that appears when $sigma in (frac{1}{sqrt{2}},sigma^*)$, for a certain $sigma^* in left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right]$. Notice that in the present model the admitted nonlinearities for which asymptotic stability of solitons is proved, are subcritical in the sense that it does not give rise to blow up, regardless of the chosen initial data. The second model is the energy critical focusing nonlinear Schrödinger equation $i frac{du}{dt}=-Delta u-|u|^4 u$. In this case we prove, for any $nu$ and $alpha_0$ sufficiently small, the existence of radial finite energy solutions of the form$u(t,x)=e^{ialpha(t)}lambda^{1/2}(t)W(lambda(t)x)+e^{iDeltat}zeta^*+o_{dot H^1} (1)$ as $tright arrow +infty$, where$alpha(t)=alpha_0ln t$, $lambda(t)=t^{nu}$,$W(x)=(1+frac13|x|^2)^{-1/2}$ is the ground state and $zeta^*$is arbitrarily small in $dot H^1$