Wigner Functions versus WKB-Methods in Multivalued Geometrical Optics

We consider the Cauchy-problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of high-frequency asymptotics of such models is reviewed,in particular we highlight the difficulties in crossing caustics when using (time-dependent) WKB-methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono-kinetic solutions) in the prebreaking regime. Further we show that the Wigner measure approach can be used to analyze high-frequency limits in the post-breaking regime, in comparison with the traditional Fourier integral operator method. Finally we present some illustrating examples.Comment: 38 page

On the Long Time Behavior of the Quantum Fokker-Planck equation

We analyze the long time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type scattering operator, subject to confining potentials of harmonic oscillator type. We establish the conditions under which there exists a thermal equilibrium state and prove exponential decay towards it, using (classical) entropy-methods. Additionally, we give precise dispersion estimates in the cases were no equilibrium state exists

Optimal bilinear control of Gross-Pitaevskii equations

A mathematical framework for optimal bilinear control of nonlinear Schr\"odinger equations of Gross-Pitaevskii type 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. Well-posedness of the problem and existence of an optimal control is proven. 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.Comment: 30 pages, 14 figure

A time-splitting spectral scheme for the Maxwell-Dirac system

We present a time-splitting spectral scheme for the Maxwell-Dirac system and similar time-splitting methods for the corresponding asymptotic problems in the semi-classical and the non-relativistic regimes. The scheme for the Maxwell-Dirac system conserves the Lorentz gauge condition, is unconditionally stable and highly efficient as our numerical examples show. In particular we focus in our examples on the creation of positronic modes in the semi-classical regime and on the electron-positron interaction in the non-relativistic regime. Furthermore, in the non-relativistic regime, our numerical method exhibits uniform convergence in the small parameter \dt, which is the ratio of the characteristic speed and the speed of light.Comment: 29 pages, 119 figure

Classical limit for semi-relativistic Hartree systems

We consider the three-dimensional semi-relativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove, using Wigner transformation techniques, that its classical limit yields the well known relativistic Vlasov-Poisson system. The result holds for the case of attractive and repulsive mean-field interaction, with an additional size constraint in the attractive case.Comment: 10 page

Ground Support Equipment for Northrop Grumman Massive Heat Transfer Experiment

California Polytechnic State University students designed, built, and certified ground support equipment for the Northrop Grumman Massive Heat Transfer Experiment. The Cal Poly design team built the 10000, 20000, and 30000 assemblies to meet Northrop Grumman requirements. The requirements included interface limitations, design load factors, delivery, and testing specifications. The design process consists of requirements generation, conceptual design, preliminary design, design reviews, manufacturing, and certification. The hardware was successfully completed and is used at the Johnson Space and Kennedy Space Center

Quantum dynamical semigroups for diffusion models with Hartree interaction

We consider a class of evolution equations in Lindblad form, which model the dynamics of dissipative quantum mechanical systems with mean-field interaction. Particularly, this class includes the so-called Quantum Fokker-Planck-Poisson model. The existence and uniqueness of global-in-time, mass preserving solutions is proved, thus establishing the existence of a nonlinear conservative quantum dynamical semigroup. The mathematical difficulties stem from combining an unbounded Lindblad generator with the Hartree nonlinearity.Comment: 30 pages; Introduction changed, title changed, easier and shorter proofs due to new energy norm. to appear in Comm. Math. Phy

Determining the probability of cyanobacterial blooms: the application of Bayesian networks in multiple lake systems

A Bayesian network model was developed to assess the combined influence of nutrient conditions and climate on the occurrence of cyanobacterial blooms within lakes of diverse hydrology and nutrient supply. Physicochemical, biological, and meteorological observations were collated from 20 lakes located at different latitudes and characterized by a range of sizes and trophic states. Using these data, we built a Bayesian network to (1) analyze the sensitivity of cyanobacterial bloom development to different environmental factors and (2) determine the probability that cyanobacterial blooms would occur. Blooms were classified in three categories of hazard (low, moderate, and high) based on cell abundances. The most important factors determining cyanobacterial bloom occurrence were water temperature, nutrient availability, and the ratio of mixing depth to euphotic depth. The probability of cyanobacterial blooms was evaluated under different combinations of total phosphorus and water temperature. The Bayesian network was then applied to quantify the probability of blooms under a future climate warming scenario. The probability of the "high hazardous" category of cyanobacterial blooms increased 5% in response to either an increase in water temperature of 0.8°C (initial water temperature above 24°C) or an increase in total phosphorus from 0.01 mg/L to 0.02 mg/L. Mesotrophic lakes were particularly vulnerable to warming. Reducing nutrient concentrations counteracts the increased cyanobacterial risk associated with higher temperatures

Multiphase weakly nonlinear geometric optics for Schrodinger equations

We describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrodinger equations, posed on Euclidean space or on the torus. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution, but does not alter the rapid oscillations. We consider initial states which are superpositions of slowly modulated plane waves, and use the framework of Wiener algebras. A detailed analysis of the corresponding nonlinear wave mixing phenomena is given, including a geometric interpretation on the resonance structure for cubic nonlinearities. As an application, we recover and extend some instability results for the nonlinear Schrodinger equation on the torus in negative order Sobolev spaces.Comment: 29 page