In this diploma work, two versions of the discrete nonlinear Schrödinger (DNLS) equation are used to model a nonlinear layered photonic crystal system; the cubic DNLS (cDNLS) equation and the saturable DNLS (sDNLS) equation. They both have site-dependent coefficients to break mirror symmetry with respect to propagation direction, as well as to describe the linear and nonlinear properties of the system. Analytical solutions taking on plane wave form are, via the backward transfer map, used to derive a transmission coefficient as well as a rectifying factor to quantify the diode effect. The effect of varying site-dependent coefficients is studied in detail. Numerical simulations of Gaussian wave packets impinging on the system, using open boundary conditions, show the breaking of parity symmetry. Evidence of a change in the wave packet dynamics occurring in the transition between the cubic and the saturable DNLS model is presented. A saturated system prevents the wave packet from getting stuck in the nonlinear lattice layers. The transmission properties were found to be very sensitive to slight changes of the system parameters.

We study ground states in the nonlinear Schrödinger equation (NLS) with an isotropic harmonic potential, in energy-critical and energy-supercritical cases. In both cases, we prove existence of a family of ground states parametrized by their amplitude, together with the corresponding values of the spectral parameter. Moreover, we derive asymptotic behavior of the spectral parameter when the amplitude of ground states tends to infinity. We show that in the energy-supercritical case the family of ground states converges to a limiting singular solution and the spectral parameter converges to a nonzero limit, where the convergence is oscillatory for smaller dimensions, and monotone for larger dimensions. In the energy-critical case, we show that the spectral parameter converges to zero, with a specific leading-order term behavior depending on the spatial dimension. Furthermore, we study the Morse index of the ground states in the energy-supercritical case. We show that in the case of monotone behavior of the spectral parameter, that is for large values of the dimension, the Morse index of the ground state is finite and independent of its amplitude. Moreover, we show that it asymptotically equals to the Morse index of the limiting singular solution. This result suggests how to estimate the Morse index of the ground state numerically. / Dissertation / Doctor of Philosophy (PhD)

We clear up two issues regarding the eigenvalue problem for the Manakov system; these problems relate directly to the existence of the soliton [sic] effect in fiber optic cables. The first issue is a bound on the eigenvalues of the Manakov system: if the parameter ξ is an eigenvalue, then it must lie in a certain region in the complex plane. The second issue has to do with a chirped Manakov system. We show that if a system is chirped too much, the soliton effect disappears. While this has been known for some time experimentally, there has not yet been a theoretical result along these lines for the Manakov system. / Ph. D.

The purpose of this thesis is to study the effects of gain and damping on a nonlinear waveguide array with a strong disorder that is driven in the first site, and try to find regimes which have stable stationary solutions. This has been done with a modified DNLS (Discrete nonlinear Schrödinger equation). Stable stationary solutions were mainly found when the damping was stronger than the gain, but some stable stationary regimes were also found when the gain was stronger than the damping.

This thesis deals with problems arising in the study of nonlinear partial differential equations arising from many-body problems. It is divided into two parts: The first part concerns the derivation of a nonlinear diffusion equation from a microscopic stochastic process. We give a new method to show that in the hydrodynamic limit, the particle densities of a one-dimensional zero range process on a periodic lattice converge to the solution of a nonlinear diffusion equation. This method allows for the first time an explicit uniform-in-time bound on the rate of convergence in the hydrodynamic limit. We also discuss how to extend this method to the multi-dimensional case. Furthermore we present an argument, which seems to be new in the context of hydrodynamic limits, how to deduce the convergence of the microscopic entropy and Fisher information towards the corresponding macroscopic quantities from the validity of the hydrodynamic limit and the initial convergence of the entropy. The second part deals with problems arising in the analysis of nonlinear Schrödinger equations of Gross-Pitaevskii type. First, we consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in the literature. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case. Finally, a mathematical framework for optimal bilinear control of nonlinear Schrödinger equations arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used L^2- or H^1-norms for the cost of the control action. We prove well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.

This thesis is devoted to the study of excitations in atomic and polariton Bose-Einstein condensates (BEC). These two specimens are prime examples for equilibrium and non equilibrium BEC. The corresponding condensate wave function of each system satisfies a particular partial differential equation (PDE). These PDEs are discussed in the beginning of this thesis and justified in the context of the quantum many-body problem. For high occupation numbers and when neglecting quantum fluctuations the quantum field operator simplifies to a semiclassical wave. It turns out that the interparticle interactions can be simplified to a single parameter, the scattering length, which gives rise to an effective potential and introduces a nonlinearity to the PDE. In both cases, i.e. equilibrium and non equilibrium, the main model corresponding to the semiclassical wave is the Gross-Pitaevskii equation (GPE), which includes certain mathematical adaptions depending on the physical context of the consideration and the nature of particles/quasiparticles, such as additional complex pumping and growth terms or terms due to motion. In the course of this work I apply a variety of state-of-the-art analytical and numerical tools to gain information about these semiclassical waves. The analytical tools allow e.g. to determine the position of the maximum density of the condensate wave function or to find the critical velocities at which excitations are expected to be generated within the condensate. In addition to analytical considerations I approximate the GPE numerically. This allows to gain the condensate wave function explicitly and is often a convenient tool to study the emergence of excitations in BEC. It is in particular shown that the form of the possible excitations significantly depends on the dimensionality of the considered system. The generated excitations within the BEC include quantum vortices, quantum vortex rings or solitons. In addition multicomponent systems are considered, which enable more complex dynamical scenarios. Under certain conditions imposed on the condensate one obtains dark-bright soliton trains within the condensate wave function. This is shown numerically and analytical expressions are found as well. In the end of this thesis I present results as part of an collaborative effort with a group of experimenters. Here it is shown that the wave function due to a complex GPE fits well with experiments made on polariton condensates, statically and dynamically.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22863号 / 情博第742号 / 新制||情||127(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 矢ヶ崎 一幸, 教授 中村 佳正, 准教授 柴山 允瑠, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-07-17T13:07:11Z No. of bitstreams: 2 Tese - Luciano Calaça Alves - 2018.pdf: 4699371 bytes, checksum: 706846314ebb74648be890b03e18d4cc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-07-17T13:55:46Z (GMT) No. of bitstreams: 2 Tese - Luciano Calaça Alves - 2018.pdf: 4699371 bytes, checksum: 706846314ebb74648be890b03e18d4cc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-07-17T13:55:46Z (GMT). No. of bitstreams: 2 Tese - Luciano Calaça Alves - 2018.pdf: 4699371 bytes, checksum: 706846314ebb74648be890b03e18d4cc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-06-07 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / This work presents the study of solitary wave solutions, known as solitons, in non-linear and non- homogeneous media using non-linear Schrödinger equations. Three cases are studied: first considering a logarithmic nonlinear term; second with saturation effect and finally including effects of high orders (Raman scattering). Solutions are modulated by three different types of potential. First, linear in the spatial and oscillatory coordinate in the temporal coordinate. The second, quadratic in the spatial and oscillatory in the temporal coordinates. Finally, it is also modulated using a mixed potential, which is the junction of the two potentials presented above. After including inomogeneities in linear and nonlinear coefficients, the similarity transformation technique is used to convert the non-linear, non-autonomous equation into an autonomous one that will be solved analytically. This field of study has potential applications in crystals, optical fibers and in Bose- Einstein condensates, also serving to understand the fundamentals related to this state of matter. The stability of the solutions are checked by numerical simulations. / Este trabalho apresenta o estudo de soluções de ondas solitárias, conhecidas como sólitons, em meios não lineares e não homogêneos por meio de equações não lineares de Schrödinger. São estudados três casos: primeiro considerando um termo não linear do tipo logarítmico; segundo com efeito de saturação e por último incluindo efeitos de altas ordens (espalhamento Raman). As soluções são moduladas por três tipos diferentes de potencial. O primeiro, linear na coordenada espacial e oscilatório na coordenada temporal. O segundo, quadrático na coordenada espacial e oscilatório na temporal. Por fim, modula-se também utilizando um potencial misto, que é a junção dos dois potenciais apresentados anteriormente. Depois de incluir heterogeneidades nos coeficientes lineares e não lineares, é utilizada a técnica da transformação de similaridade para converter a equação não linear, não autônoma em uma autônoma que será resolvida analiticamente. Esse campo de estudo tem potenciais aplicações em cristais, fibras ópticas e em condensados de Bose-Einstein, servindo também para o entendimento dos fundamentos relacionados a esse estado da matéria. A estabilidade das soluções são checadas por meio de simulações numéricas.

Optical fibers play a vital role in modern telecommunication systems and networks. An optical fiber link imposes some linear and nonlinear distortions on the propagating light-wave signal due to the inherent dispersive nature and nonlinear behavior of the fiber. These distortions impede the increasing demand for higher data rate transmission over longer distances. Developing efficient and computationally non-expensive digital signal processing (DSP) techniques to effectively compensate for the fiber impairments is therefore essential and of preeminent importance. This thesis proposes two DSP-based approaches for mitigating the induced distortions in short-reach and long-haul fiber-optic communication systems. The first approach introduces a powerful digital nonlinear feed-forward equalizer (NFFE), exploiting multilayer artificial neural network (ANN). The proposed ANN-NFFE mitigates nonlinear impairments of short-haul optical fiber communication systems, arising due to the nonlinearity introduced by direct photo-detection. In a direct detection system, the detection process is nonlinear due to the fact that the photo-current is proportional to the absolute square of the electric field intensity. The proposed equalizer provides the most efficient computational cost with high equalization performance. Its performance is comparable to the benchmark compensation performance achieved by maximum-likelihood sequence estimator. The equalizer trains an ANN to act as a nonlinear filter whose impulse response removes the intersymbol interference (ISI) distortions of the optical channel. Owing to the proposed extensive training of the equalizer, it achieves the ultimate performance limit of any feed-forward equalizer. The performance and efficiency of the equalizer are investigated by applying it to various practical short-reach fiber-optic transmission system scenarios. These scenarios are extracted from practical metro/media access networks and data center applications. The obtained results show that the ANN-NFFE compensates for the received BER degradation and significantly increases the tolerance to the chromatic dispersion distortion. The second approach is devoted for blindly combating impairments of long-haul fiber-optic systems and networks. A novel adjoint sensitivity analysis (ASA) approach for the nonlinear Schrödinger equation (NLSE) is proposed. The NLSE describes the light-wave propagation in optical fiber communication systems. The proposed ASA approach significantly accelerates the sensitivity calculations in any fiber-optic design problem. Using only one extra adjoint system simulation, all the sensitivities of a general objective function with respect to all fiber design parameters are estimated. We provide a full description of the solution to the derived adjoint problem. The accuracy and efficiency of our proposed algorithm are investigated through a comparison with the accurate but computationally expensive central finite-differences (CFD) approach. Numerical simulation results show that the proposed ASA algorithm has the same accuracy as the CFD approach but with a much lower computational cost. Moreover, we propose an efficient, robust, and accelerated adaptive digital back propagation (A-DBP) method based on adjoint optimization technique. Provided that the total transmission distance is known, the proposed A-DBP algorithm blindly compensates for the linear and nonlinear distortions of point-to-point long-reach optical fiber transmission systems or multi-point optical fiber transmission networks, without knowing the launch power and channel parameters. The NLSE-based ASA approach is extended for the sensitivity analysis of general multi-span DBP model. A modified split-step Fourier scheme method is introduced to solve the adjoint problem, and a complete analysis of its computational complexity is studied. An adjoint-based optimization (ABO) technique is introduced to significantly accelerate the parameters extraction of the A-DBP. The ABO algorithm utilizes a sequential quadratic programming (SQP) technique coupled with the extended ASA algorithm to rapidly solve the A-DBP training problem and optimize the design parameters using minimum overhead of extra system simulations. Regardless of the number of A-DBP design parameters, the derivatives of the training objective function with respect to all parameters are estimated using only one extra adjoint system simulation per optimization iterate. This is contrasted with the traditional finite-difference (FD)-based optimization methods whose sensitivity analysis calculations cost per iterate scales linearly with the number of parameters. The robustness, performance, and efficiency of the proposed A-DBP algorithm are demonstrated through applying it to mitigate the distortions of a 4-span optical fiber communication system scenario. Our results show that the proposed A-DBP achieves the optimal compensation performance obtained using an ideal fine-mesh DBP scheme utilizing the correct channel parameters. Compared to A-DBPs trained using SQP algorithms based on forward, backward, and central FD approaches, the proposed ABO algorithm trains the A-DBP with 2.02 times faster than the backward/forward FD-based optimizers, and with 3.63 times faster than the more accurate CFD-based optimizer. The achieved gain further increases as the number of design parameters increases. A coarse-mesh A-DBP with less number of spans is also adopted to significantly reduce the computational complexity, achieving compensation performance higher than that obtained using the coarse-mesh DBP with full number of spans. / Thesis / Doctor of Philosophy (PhD) / This thesis proposes two powerful and computationally efficient digital signal processing (DSP)-based techniques, namely, artificial neural network nonlinear feed forward equalizer (ANN-NFFE) and adaptive digital back propagation (A-DBP) equalizer, for mitigating the induced distortions in short-reach and long-haul fiber-optic communication systems, respectively. The ANN-NFFE combats nonlinear impairments of direct-detected short-haul optical fiber communication systems, achieving compensation performance comparable to the benchmark performance obtained using maximum-likelihood sequence estimator with much lower computational cost. A novel adjoint sensitivity analysis (ASA) approach is proposed to significantly accelerate sensitivity analyses of fiber-optic design problems. The A-DBP exploits a gradient-based optimization method coupled with the ASA algorithm to blindly compensate for the distortions of coherent-detected fiber-optic communication systems and networks, utilizing the minimum possible overhead of performed system simulations. The robustness and efficiency of the proposed equalizers are demonstrated using numerical simulations of varied examples extracted from practical optical fiber communication systems scenarios.

Cette thèse étudie l'influence du vent sur l'instabilité modulationnelle. Une première partie unifie les travaux de Segur et al. qui intègrent la dissipation et ceux de Leblanc qui prennent en compte le vent. Une équation non linéaire de Schrödinger est établie avec un terme additionnel linéaire résultant de la compétition entre le vent et la dissipation. La dissipation est traduite par le modèle de Lundgren et l'effet du vent se manifeste par l'intermédiaire de la pression atmosphérique selon le modèle de Miles. La profondeur est finie. Une étude de stabilité de l'onde de Stokes est détaillée, et des simulations numériques sont menées pour illustrer les résultats. Des expérimentations sont menées pour apporter une validation qualitative à ces travaux. Cette première partie a été validée par une publication au Journal of Fluid Mechanics (2010). La deuxième partie étudie l'influence du vent sur l'instabilité modulationnelle par l'intermédiaire de la vorticité qu'il crée en surface. Le modèle est simplifié par l'hypothèse d'un écoulement unidirectionnel et d'une vorticité constante. La profondeur est encore supposée finie. Une équation non linéaire de Schrödinger est établie, qui prend en compte cette vorticité constante. La stabilité de l'onde de Stokes est alors étudiée en détail(diagramme d'instabilité en fonction de la vorticité et de la profondeur, bande d'instabilité, taux d'instabilité, etc.). Il est démontré qu'une vorticité négative, au delà d'un certain seuil, supprime l'instabilité modulationnelle indépendamment de la profondeur. Cette deuxième partie a été soumise pour publication au journal Physics of Fluids. / This thesis manuscript treats about the influence of wind on modulational instability. A first part merges the works of Segur at al. which take into account viscous dissipation and Leblanc's work which deals with wind. A nonlinear Schrödinger equation is derived, with a forcing linear term which represents the result of the balance between wind forcing and dissipation. Visous dissipation is represented by Lundgren's model and the effect of wind is integrated into atmospheric pressure following Miles' model. Depth is finite. The stability of Stokes's waves is investigated, and numerical simulations are presented to illustrate the results. Some experimentations are done to confirm qualitatively these works. This first part was validated by a publication in the Journal of Fluid Mechanics~(2010). The second part studies the influence of the wind on the modulational instability by the intermediary of the vorticity whom it creates on the water at the surface. The model is simplified by the hypothesis of an unidirectional flow and a constant vorticity. The depth is still supposed finite. A non linear Schrödinger equation is derived, which takes into account this constant vorticity. The stability of the Stokes' wave is studied then in detail (instability diagram function of vorticity and depth, instability bandwidth, instability rate, etc.). It is demonstrated that a negative vorticity, beyond a certain threshold, eliminates the modulational instability independently of the depth. This second part has been submitted for publication in the journal Physics of Fluids.