Ghosts of order on the frontier of chaos

What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderly, completely integrable systems are characterized by phase trajectories confined to low-dimensional, invariant tori. The KAM theory examines what happens to the tori when an integrable system is subjected to a small perturbation and finds that, for small enough perturbations, most of them survive.

The KAM theory is mute about the disrupted tori, but, for two-dimensional systems, Aubry and Mather discovered an astonishing picture: the broken tori are replaced by "cantori," tattered, Cantor-set remnants of the original invariant curves. We seek to extend Aubry and Mather's picture to higher dimensional systems and report two kinds of studies; both concern perturbations of a completely integrable, four-dimensional symplectic map. In the first study we compute some numerical approximations to Birkhoff periodic orbits; sequences of such orbits should approximate any higher dimensional analogs of the cantori. In the second study we prove converse KAM theorems; that is, we use a combination of analytic arguments and rigorous, machine-assisted computations to find perturbations so large that no KAM tori survive. We are able to show that the last few of our Birkhoff orbits exist in a regime where there are no tori.

Fine interference fringes formed in high-order harmonic spectra generated by infrared driving laser pulses

We experimentally investigate the high-order harmonic generation in argon gas using a driving laser pulse at a center wavelength of 1240 nm. High-contrast fine interference fringes could be observed in the harmonic spectra near the propagation axis, which is attributed to the interference between long and short quantum paths. We also systematically examine the variation of the interference fringe pattern with increasing energy of the driving pulse and with different phase-matching conditions.

Laser intensity dependence of high-order harmonic generation from aligned CO2 molecules

We investigate experimentally the high-order harmonic generation from aligned CO2 molecules and demonstrate that the modulation inversion of the harmonic yield with respect to molecular alignment can be altered dramatically by fine-tuning the intensity of the driving laser pulse for harmonic generation. The results can be modeled by employing the strong field approximation including a ground state depletion factor. The laser intensity is thus proved to be a parameter that can control the high-harmonic emission from aligned molecules.

Spectral evolution of angularly resolved third-order harmonic generation by infrared femtosecond laser-pulse filamentation in air

We experimentally investigate the evolution of an angularly resolved spectrum of third harmonic generated by infrared femtosecond laser pulse filamentation in air. We show that at low pump intensity, phase matching between the fundamental and third-harmonic waves dominates the nonlinear optical effect and induces a ring structure of the third-harmonic beam, whereas at high pump intensity, the dispersion properties of air begin to affect the angular spectrum, leading to the formation of a nonlinear X wave at third harmonic.

Tunable high-order harmonic generation and the role of the folded quantum path

We theoretically demonstrate the selective enhancement of high-order harmonic generation (HHG) in two-color laser fields consisting of a single-cycle fundamental wave (800 nm wavelength) and a multicycle subharmonic wave (2400 nm wavelength). By performing time-frequency analyses based on a single-active-electron model, we reveal that such an enhancement is a result of the modified electron trajectories in the two-color field. Furthermore, we show that selectively enhanced HHG gives rise to a bandwidth-controllable extreme ultraviolet supercontinuum in the plateau region, facilitating the generation of intense single isolated attosecond pulses.

Phase-matched high-order harmonic generation in a gas cell with midinfrared femtosecond pulses

We experimentally investigate the generation of high-order harmonics in a 4-mm-long gas cell using midinfrared femtosecond pulses at various wavelengths of 1240 nm, 1500 nm, and 1800 nm. It is observed that the yield and cutoff energy of the generated high-order harmonics critically depend on focal position, gas pressure, and size of the input beam which can be controlled by an aperture placed in front of the focal lens. By optimizing the experimental parameters, we achieve a cutoff energy at similar to 190 eV with the 1500 nm driving pulses, which is the highest for the three wavelengths chosen in our experiment.

Geometric integration applied to moving mesh methods and degenerate Lagrangians

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this thesis we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. The proposed methods are readily applicable to (weakly) non-degenerate field theories---numerical results for the Sine-Gordon equation are presented.

In an attempt to extend our approach to degenerate field theories, in the last part of this thesis we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the 'Hamiltonian' equations of motion can be formulated as an index 1 differential-algebraic system. We then proceed to construct variational Runge-Kutta methods and analyze their properties. The general properties of Runge-Kutta methods depend on the 'velocity' part of the Lagrangian. If the 'velocity' part is also linear in the position coordinate, then we show that non-partitioned variational Runge-Kutta methods are equivalent to integration of the corresponding first-order Euler-Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge-Kutta method are retained. If the 'velocity' part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We also apply our methods to several models and present the results of our numerical experiments.

Optimal guidance of low-thrust interplanetary space vehicles

The low-thrust guidance problem is defined as the minimum terminal variance (MTV) control of a space vehicle subjected to random perturbations of its trajectory. To accomplish this control task, only bounded thrust level and thrust angle deviations are allowed, and these must be calculated based solely on the information gained from noisy, partial observations of the state. In order to establish the validity of various approximations, the problem is first investigated under the idealized conditions of perfect state information and negligible dynamic errors. To check each approximate model, an algorithm is developed to facilitate the computation of the open loop trajectories for the nonlinear bang-bang system. Using the results of this phase in conjunction with the Ornstein-Uhlenbeck process as a model for the random inputs to the system, the MTV guidance problem is reformulated as a stochastic, bang-bang, optimal control problem. Since a complete analytic solution seems to be unattainable, asymptotic solutions are developed by numerical methods. However, it is shown analytically that a Kalman filter in cascade with an appropriate nonlinear MTV controller is an optimal configuration. The resulting system is simulated using the Monte Carlo technique and is compared to other guidance schemes of current interest.

Optimization of high-order harmonic by genetic algorithm for the chirp and phase of few-cycle pulses

The brightness of a particular harmonic order is optimized for the chirp and initial phase of the laser pulse by genetic algorithm. The influences of the chirp and initial phase of the excitation pulse on the harmonic spectra are discussed in terms of the semi-classical model including the propagation effects. The results indicate that the harmonic intensity and cutoff have strong dependence on the chirp of the laser pulse, but slightly on its initial phase. The high-order harmonics can be enhanced by the optimal laser pulse and its cutoff can be tuned by optimization of the chirp and initial phase of the laser pulse.

Multiscale model reduction methods for deterministic and stochastic partial differential equations

Partial differential equations (PDEs) with multiscale coefficients are very difficult to solve due to the wide range of scales in the solutions. In the thesis, we propose some efficient numerical methods for both deterministic and stochastic PDEs based on the model reduction technique.

For the deterministic PDEs, the main purpose of our method is to derive an effective equation for the multiscale problem. An essential ingredient is to decompose the harmonic coordinate into a smooth part and a highly oscillatory part of which the magnitude is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is smooth, and could be resolved on a regular coarse mesh grid. Furthermore, we provide error analysis and show that the solution to the effective equation plus a correction term is close to the original multiscale solution.

For the stochastic PDEs, we propose the model reduction based data-driven stochastic method and multilevel Monte Carlo method. In the multiquery, setting and on the assumption that the ratio of the smallest scale and largest scale is not too small, we propose the multiscale data-driven stochastic method. We construct a data-driven stochastic basis and solve the coupled deterministic PDEs to obtain the solutions. For the tougher problems, we propose the multiscale multilevel Monte Carlo method. We apply the multilevel scheme to the effective equations and assemble the stiffness matrices efficiently on each coarse mesh grid. In both methods, the $\KL$ expansion plays an important role in extracting the main parts of some stochastic quantities.

For both the deterministic and stochastic PDEs, numerical results are presented to demonstrate the accuracy and robustness of the methods. We also show the computational time cost reduction in the numerical examples.

Ortho-positronium annihilation: steps toward computing the first order radiative corrections

The 0.2% experimental accuracy of the 1968 Beers and Hughes measurement of the annihilation lifetime of ortho-positronium motivates the attempt to compute the first order quantum electrodynamic corrections to this lifetime. The theoretical problems arising in this computation are here studied in detail up to the point of preparing the necessary computer programs and using them to carry out some of the less demanding steps -- but the computation has not yet been completed. Analytic evaluation of the contributing Feynman diagrams is superior to numerical evaluation, and for this process can be carried out with the aid of the Reduce algebra manipulation computer program.

The relation of the positronium decay rate to the electronpositron annihilation-in-flight amplitude is derived in detail, and it is shown that at threshold annihilation-in-flight, Coulomb divergences appear while infrared divergences vanish. The threshold Coulomb divergences in the amplitude cancel against like divergences in the modulating continuum wave function.

Using the lowest order diagrams of electron-positron annihilation into three photons as a test case, various pitfalls of computer algebraic manipulation are discussed along with ways of avoiding them. The computer manipulation of artificial polynomial expressions is preferable to the direct treatment of rational expressions, even though redundant variables may have to be introduced.

Special properties of the contributing Feynman diagrams are discussed, including the need to restore gauge invariance to the sum of the virtual photon-photon scattering box diagrams by means of a finite subtraction.

A systematic approach to the Feynman-Brown method of Decomposition of single loop diagram integrals with spin-related tensor numerators is developed in detail. This approach allows the Feynman-Brown method to be straightforwardly programmed in the Reduce algebra manipulation language.

The fundamental integrals needed in the wake of the application of the Feynman-Brown decomposition are exhibited and the methods which were used to evaluate them -- primarily dis persion techniques are briefly discussed.

Finally, it is pointed out that while the techniques discussed have permitted the computation of a fair number of the simpler integrals and diagrams contributing to the first order correction of the ortho-positronium annihilation rate, further progress with the more complicated diagrams and with the evaluation of traces is heavily contingent on obtaining access to adequate computer time and core capacity.