Cette thèse porte sur l'étude des phénomènes d'explosion pour certaines équations aux dérivées partielles dispersives et plus particulièrement pour l'équation de Schrodinger non linéaire. Ces phénomènes ont été beaucoup étudiés et notamment dans le cas Euclidien. On s'intéresse ici à des cas où l'espace n'est plus l'espace Euclidien. Cela comprend en particulier l'étude des trois prototypes : domaine de l'espace Euclidien, tore (courbure nulle), sphère (courbure positive) et espace hyperbolique (courbure négative). Concernant l'équation de Schrodinger, plusieurs résultats ont montré que la métrique pouvait influencer le comportement qualitatif des solutions, en particulier les propriétés dispersives des solutions et le seuil critique d'existence locale pour le problème de Cauchy. Plusieurs résultats concernant l'explosion sont ensuite venus confirmer ces phénomèmes. Dans cette thèse, on se propose de poursuivre cette étude. / In this thesis, we study blow-up behavior of solutions for dispersive equations, more precisely for the nonlinear Schr"odinger equation. This has been studied essentially in the Euclidean case. In this work, we are interested in the case where the equation is posed on a general manifold; this includes the case of a domain of the Euclidean space, torus (zero curvature); the sphere (non negative curvature) and the hyperbolic space (negative curvature). For the Schr"odinger equation, several results proved that the metric could change the qualitative behavior of the solutions, in particular dispersive properties and the critical threshold of existence for the Cauchy problem. Then, some results showed that blow-up theory is also concerned. In this work, we continue this study.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18768号 / 理博第4026号 / 新制||理||1580(附属図書館) / 31719 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 堤 誉志雄, 教授 上田 哲生, 教授 加藤 毅 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20152号 / 理博第4237号 / 新制||理||1609(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 堤 誉志雄, 教授 上田 哲生, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The dissertation focuses on the nonlinear Schrodinger equation iu_t+u_{xx}+kappa|u|^2u =0, for the complex-valued function u=u(x,t) with domain t>=0, 0<=x<= L, where the parameter kappa is any non-zero real number. It is shown that the problem is locally and globally well-posed for appropriate initial data and the solution exponentially decays to zero as t goes to infinity under the boundary conditions u(0,t) = beta u(L,t) and beta u_x(0,t)-u_x(L,t) = ialpha u(0,t), where L>0, and alpha and beta are any real numbers satisfying alpha*beta<0 and beta does not equal 1 or -1. Moreover, the numerical study of controllability problem for the nonlinear Schrodinger equations is given. It is proved that the finite-difference scheme for the linear Schrodinger equation is uniformly boundary controllable and the boundary controls converge as the step sizes approach to zero. It is then shown that the discrete version of the nonlinear case is boundary null-controllable by applying the fixed point method. From the new results, some open questions are presented. / Ph. D. / The dissertation concerns the solutions of nonlinear Schrodinger (NLS) equation, which arises in many applications of physics and applied mathematics and models the propagation of light waves in fiber optics cables, surface water-waves, Langmuir waves in a hot plasma, oceanic and optical rogue waves, etc. Under certain dissipative boundary conditions, it is shown that for given initial data, the solutions of NLS equation always exist for a finite time, and for small initial data, the solutions exist for all the time and decay exponentially to zero as time goes to infinity. Moreover, by applying a boundary control at one end of the boundary, it is shown using a finite-difference approximation scheme that the linear Schrodinger equation is uniformly controllable. It is proved using fixed point method that the discrete version of the NLS equation is also boundary controllable. The results obtained may be applicable to design boundary controls to eliminate unwanted waves generated by noises as well as create the wave propagation that is important in applications.

Orientador: Marcelo Batista Hott / Coorientador: Alvaro de Souza Dutra / Banca: Denis Dalmazi / Banca: Roberto André Kraenkei / Banca: Othon Cabo Winter / Wesley Bueno Cardoso / Resumo: Nesta tese lidamos com a equação de Schroedinger não linear com coeficientes modulados em diferentes contextos. Esta equação diferencial não linear é amplamente usada para descrever a propagação de pulsos de luz através de uma fibra óptica ou para modelar a dinâmica de um condensado de Bose-Einstein. Primeiro, aplicamos as transformações canônicas de ponto para resolver algumas classes de equação de Schroedinger não linear com coeficientes modulados ou seja, aqueles que possuem não linearidades cúbica e quântica (dependentes do espaço e tempo) específicas. O método aplicado aqui nos permite encontrar soluções tipo sólitons localizados (no espaço) para a equação de Schroedinger não linear com coeficientes modulados, que não foram apresentados antes. No contexto de condensados de Bose-Einstein, nós generalizamos o potencial externo o qual armadilha o sistema, e os termos de não linearidade da equação diferencial. Em seguida, aplicamos as transformações canônicas de ponto para resolver algumas classes de duas equações de Schroedinger não lineares acopladas com coeficientes modula-dos isto é, não linearidades cúbica e quântica - dependentes do espaço e tempo - específicas. O método aplicado aqui nos permite encontrar uma classe de soluções de sólitons tipo vetoriais localizados (no espaço) das duas equações de Schroedinger não linear acopladas. Os sólitons vetoriais encontrados aqui podem ser aplicados a estudos teóricos de condensados de Bose-Einstein de átomos com dois estados internos diferentes ou á propagação de pulsos de luz através de fibras ópticas focalizadoras ou desfocalizadoras. Finalmente, usando transformações canônicas de ponto obtemos soluções exatas localizadas (no espaço) da equação de Schroedinger não linear com não linearidades cúbica e quântica moduladas no espaço e tempo ...(Resumo completo, clicar acesso eletrônico abaixo) / Abstract: In this thesis we deal with the nonlinear Schrödinger equation with modulated coefficients in different contexts. This nonlinear differential equation is widely used to describe light pulses propagating through an optical fiber or to model the dynamics of a Bose-Einstein condensate. First, we apply point canonical transformations to solve some classes of nonlinear Schrödinger equation with modulated coefficients namely, those which possess specific cubic and quantic (time- and space-dependent) nonlinearities. The method applied here allows us to find wide localized (in space) soliton solutions to the nonlinear Schrödinger equation, which were not presented before. In the context of Bose-Einstein condensates, we also generalize the external potential which traps the system and the nonlinearities terms. Then, we apply point canonical transformations to solve some classes of two coupled nonlinear Schrödinger equations with modulated coefficients namely, specific cubic and quantic - time and space dependent - nonlinearities. The method applied here allows us to find a class of wide localized (in space) vector soliton solutions of two coupled nonlinear Schrödinger equations. The vector solitons found here can be applied to theoretical studies of Bose-condensed atoms in two different internal states and of ultrashort pulse propagation in optical fibers with focusing and defocusing nonlinearities. Finally, we use point canonical transformations to obtain localized (in space) exact solutions of the nonlinear Schrödinger equation with cubic and quantic space and time modulated nonlinearities and in the presence of time-dependent and inhomogeneous external potentials and amplification or absorption (source or drain) term. We obtain a class of wide localized exact solutions of nonlinear Schrödinger equation in the presence of a number of non-Hermitian ... (Complete abstract click electronic access below) / Doutor

Dirhodium complexes such as carboxylates and carboxylamidates are very efficient metal catalysts used in the synthesis of pharmaceuticals and agrochemicals. Recent experimental work has indicated that there are significant differences in the isomeric ratios obtained among the possible products when synthesizing these complexes. The relative stabilities of the Rh2(NPhCOCH3)4 tolunitrile complexes, Rh2(NPhCOCH3)4(NCC6H4CH3)2, were determined at the HF/LANL2DZ ECP, 6-31G and DFT/B3LYP/LANL2DZ ECP, 6-31G levels of theory using NWChem 6.3. The LANL2DZ ECP (effective core potential) basis set was used for the rhodium atoms and 6-31G basis set was used for all other atoms. Specifically, the o-tolunitrile, m-tolunitrile, and p-tolunitrile complexes of the 2,2-trans and the 4,0- isomers of Rh2(NPhCOCH3)4 were compared.

The experiment performed by Itano, Heinzen, Bollinger and Wineland on the quantum Zeno effect is analyzed in detail through a quantum map derived by conventional quantum mechanics based on the Schrodinger equation. The analysis shows that a slight modification of their experiment leads to a significantly different result from the one that is predicted through von Neumann's hypothesis of collapse of the wave function in the quantum measurement theory. This may offer a possibility of an experimental test of von Neumann's quantum measurement theory. / text

An expansion method for the stationary Schrodinger equation of a particle moving freely in an arbitrary axisymmeric three dimensional region defined by an analytic function is introduced. The region is transformed into the unit ball by means of coordinate substitution. As a result the Schrodinger equation is considerably changed. The wavefunction is expanded into a series of spherical harmonics, thus, reducing the transformed partial differential equation to an infinite system of coupled ordinary differential equations. A Fourier-Bessel expansion of the solution vector in terms of Bessel functions with real orders is employed, resulting in a generalized matrix eigenvalue problem. The method is applied to two particular examples. The first example is a prolate spheroidal billiard which is also treated by using an alternative method. The numerical results obtained by using both the methods are compared. The second exampleis a billiard family depending on a parameter. Numerical results concerning the second example include the statistical analysis of the eigenvalues.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--Stellenbosch University, 2008. / The NLS is a universal equation of the class of nonlinear integrable systems. The aim of this thesis is to study the NLS numerically. More speci cally, an algorithm is developed to calculate its nonlinear spectrum. The nonlinear spectrum is then used as a diagnostic for numerical studies of the NLS. The spectrum consists of a discrete part, further subdivided into the main part, the auxiliary part, and the continuous spectrum. Two algorithms are developed for calculating the main spectrum. One is based on Floquet theory, rst implemented by Overman [12]. The other is a direct calculation of the eigenvalues by Herbst and Weideman [16]. These algorithms are combined through the marching squares algorithm to calculate the continuous spectrum. All ideas are illustrated by numerical examples.

Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009. / In this thesis we aim to show that the Schr odinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian. After a brief introduction into non-Hermiticity, we will focus on solving the Schr odinger equation with a special class of non-Hermitian Hamiltonians, namely PT - symmetric Hamiltonians. PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing speci cally on obtaining the real and discrete energy spectrum. Various methods for solving this problematic Schr odinger equation will be considered. After starting with perturbation theory, we will move on to numerical methods. Three di erent categories of methods will be discussed. First there is the shooting method based on a Runge-Kutta solver. Next, we investigate various implementations of the spectral method. Finally, we will look at the Riccati-Pad e method, which is a numerical implemented analytical method. PT -symmetric potentials need to be solved along a contour in the complex plane. We will propose modi cations to the numerical methods to handle this. After solving the widely documented PT -symmetric Hamiltonian H = p2 􀀀(ix)N with these methods, we give a discussion and comparison of the obtained results. Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their in uence on the results.