The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2-norm concentration on a ring, bounded H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data for which the corresponding evolution by the cubic nonlinear Schrodinger equation explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.

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In this thesis, the spectral behavior of the fundamental and sum-frequency waves, generated from the surface of a thin metal film in the Kretschmann configuration, is theoretically studied with coherent ultrashort pulses. As a first exploration of considering spectral response in nonlinear plasmonics, it is shown that the spectra of reflected sum-frequency waves exhibit pronounced shifts for the incident fundamental waves close to the plasmon coupling angle, whereas meanwhile those of reflected fundamental waves display energy holes. We also demonstrate that the scale of discovered plasmon-enhanced spectral changes is strongly influenced by the magnitude of the incidentce angle and the source pulse duration, and at a certain angle a spectral switch is observed. The appearance of large sum-frequency wave shifts can serve as an unambiguous plasmon signatur in nonlinear surface spectroscopy. Also, the discovered spectral switch can trigger extremely surface-sensitive nonlinear plasmonic sensors.

Columns subjected to time varying axial load may exhibit dynamic instability due to parametric resonance. This type of instability is inherent in structures; it is not due to material or geometrical imperfections, and can occur even in perfectly constructed structures. This characteristic makes parametric resonance a very difficult to predict and therefore dangerous phenomenon. In this thesis the stability of a structural column under bounded noise axial load is studied by use of Lyapunov exponents. Bounded noise is especially useful as a loading because it may be used to represent both wide and narrow band processes, making the stability equations developed general enough to handle a wide variety of real world probabilistic loadings. The equation of motion of the first mode of vibration for this system is a second-order nonlinear stochastic ordinary differential equation. The nonlinearity makes the system exhibit bifurcating behaviour where stability shifts from the trivial solution to a non-zero mean stationary solution. The stability of the trivial and non-trivial solutions is important in obtaining a complete picture of the dynamical behaviour of the system. The effect that damping, the amplitude of noise, and the level of nonlinearity have on the stability of a structural column is studied using both analytical and numerical approaches. The largest Lyapunov exponent of the trivial solution is determined analytically by using time averaged versions of the original equation of motion. The validity of the analytical time averaged equation of motion is also verified with Monte Carlo simulations. Due to the mathematical complexity the largest Lyapunov exponent of the non-trivial stationary solutions is obtained using Monte Carlo simulation only.

Interference has long been used for precision measurement of path-length changes. Since the advent of the laser, interference has become one of the most versatile tools in metrology. Specifically, ultra-short laser pulses allow unprecedented resolution in absolute length measurements. While ultra-short laser pulses lead to high resolution, for example in white-light interferometry, they are very susceptible to dispersion. Quantum resources have been proposed to overcome some of the problems related to distortions in the interferometric signal. For example, the Hong-Ou-Mandel (HOM) interferometer relies on frequency-entangled photon pairs and features automatic even-order dispersion cancellation and high interference visibility resilient to unbalanced loss. Quantum-OCT is a technique based on HOM interferometry, that promises to overcome Optical Coherence Tomography (OCT) a classical imaging technique based on low coherence light. Furthermore, straightforward modifications of the HOM interferometer can display several different interferometric signals, including the HOM peak, quantum beating, and phase super-resolution. However, the quantum resources required are hard to produce and dim, leading to long integration times and single-photon counting. In this thesis, we introduce the theory behind Chirped-Pulse Interferometry (CPI), a new technique that combines all the advantages of Q-OCT, including even-order dispersion cancellation, but without the need for any quantum resources. We then experimentally implement CPI and demonstrate all the important characteristics shared by the HOM interferometer, but at dramatically larger signal levels. We show how CPI can be used to measure dispersion cancelled axial profiles of an optical sample and show the improvement in resolution over white-light interferometry. Finally, we show that by modifying CPI in analogous ways to HOM, CPI can also be made to produce interferometric signal identical to the HOM peak, quantum beating, and phase super-resolution.

A model is developed for the Quanser torsion system available at Control Systems Research Laboratory at Chulalongkorn University. The torsion system is a laboratory equipment that is designed for the study of position control. It consists of a DC motor that drives three inertial loads that are coupled in series with the motor, and where all components are coupled to each other through torsional springs. Several nonlinearities are observed and the most significant one is an offset in the input signal, which is compensated for. Experiments are carried out under feedback as the system is marginally stable. Different input signals are tested and used for system identification. Linear black-box state-space models are then identified using PEM, N4SID and a subspace method made for closed-loop identification, where the last two are the most successful ones. PEM is used in a second step and successfully enhances the parameter estimates from the other algorithms.

Columns subjected to time varying axial load may exhibit dynamic instability due to parametric resonance. This type of instability is inherent in structures; it is not due to material or geometrical imperfections, and can occur even in perfectly constructed structures. This characteristic makes parametric resonance a very difficult to predict and therefore dangerous phenomenon. In this thesis the stability of a structural column under bounded noise axial load is studied by use of Lyapunov exponents. Bounded noise is especially useful as a loading because it may be used to represent both wide and narrow band processes, making the stability equations developed general enough to handle a wide variety of real world probabilistic loadings. The equation of motion of the first mode of vibration for this system is a second-order nonlinear stochastic ordinary differential equation. The nonlinearity makes the system exhibit bifurcating behaviour where stability shifts from the trivial solution to a non-zero mean stationary solution. The stability of the trivial and non-trivial solutions is important in obtaining a complete picture of the dynamical behaviour of the system. The effect that damping, the amplitude of noise, and the level of nonlinearity have on the stability of a structural column is studied using both analytical and numerical approaches. The largest Lyapunov exponent of the trivial solution is determined analytically by using time averaged versions of the original equation of motion. The validity of the analytical time averaged equation of motion is also verified with Monte Carlo simulations. Due to the mathematical complexity the largest Lyapunov exponent of the non-trivial stationary solutions is obtained using Monte Carlo simulation only.

Interference has long been used for precision measurement of path-length changes. Since the advent of the laser, interference has become one of the most versatile tools in metrology. Specifically, ultra-short laser pulses allow unprecedented resolution in absolute length measurements. While ultra-short laser pulses lead to high resolution, for example in white-light interferometry, they are very susceptible to dispersion. Quantum resources have been proposed to overcome some of the problems related to distortions in the interferometric signal. For example, the Hong-Ou-Mandel (HOM) interferometer relies on frequency-entangled photon pairs and features automatic even-order dispersion cancellation and high interference visibility resilient to unbalanced loss. Quantum-OCT is a technique based on HOM interferometry, that promises to overcome Optical Coherence Tomography (OCT) a classical imaging technique based on low coherence light. Furthermore, straightforward modifications of the HOM interferometer can display several different interferometric signals, including the HOM peak, quantum beating, and phase super-resolution. However, the quantum resources required are hard to produce and dim, leading to long integration times and single-photon counting. In this thesis, we introduce the theory behind Chirped-Pulse Interferometry (CPI), a new technique that combines all the advantages of Q-OCT, including even-order dispersion cancellation, but without the need for any quantum resources. We then experimentally implement CPI and demonstrate all the important characteristics shared by the HOM interferometer, but at dramatically larger signal levels. We show how CPI can be used to measure dispersion cancelled axial profiles of an optical sample and show the improvement in resolution over white-light interferometry. Finally, we show that by modifying CPI in analogous ways to HOM, CPI can also be made to produce interferometric signal identical to the HOM peak, quantum beating, and phase super-resolution.

Foundation performance is controlled significantly by the stress-strain behavior of the underlying soils. For geomaterials, the small-strain shear modulus Gmax is a fundamental stiffness applicable to both monotonic static and dynamic loading conditions, as well to both drained and undrained loading. Yet, Gmax is too stiff for direct use in computing foundation displacements. The main objectives of this research are to: (1) explore the scaled parallelism between the stress-strain-strength behavior of the single soil element response and the load-displacement-capacity of a shallow foundation system supported on soil; (2) develop a methodology for evaluating the performance of vertically-loaded footings using a rational framework based on the small-strain modulus Gmax, large-strain strength ( and #964;max or su) and strain at failure ( and #947;f); and (3) calibrate the proposed method using a foundation database of full-scale load tests under both undrained and drained conditions. In geotechnical practice, foundation bearing capacity is handled as a limit plasticity calculation, while footing displacements are evaluated separately via elastic continuum solutions. Herein, a hybrid approach is derived that combines these two facets into a closed-form analytical solution for vertical load-deflection-capacity based on numerical studies. Here, a non-linear elastic-plastic soil model was developed to simulate the stress-strain-strength curves for simple shearing mode (LOGNEP) for each soil element. The model was encoded into a subroutine within the finite difference program FLAC. A large mesh was used to generate load-displacement curves under circular and strip footings for undrained and drained loading conditions. With proper normalization, parametric foundation response curves were generated for a variety of initial stiffnesses, shear strengths, and degrees of non-linearity in the soil stress-strain-strength response. Soil stress-strain non-linearity is described by a logarithmic function (Puzrin and Burland, 1996, 1998) that utilizes a normalized strain xL that relates strain at failure and #947;f, shear strength ( and #964;max or su), and small-strain stiffness Gmax, all having physical meaning. A closed-form algorithm is proposed for generating non-linear load-displacement curves for footings and mats within an equivalent elastic framework. The proposed method was calibrated using a database of well-documented footing load tests where soil input parameters were available from laboratory and/or in-situ field test results.

In the first part of this dissertation we explore the consequences of high frequency operation of Josephson junction arrays. At high frequencies these systems are no longer well modeled by Kirchhoffs laws, and new dynamical equations are derived directly from Maxwells equations. From these equations we derive a reduced set of averaged equations which greatly simplify the analysis of high frequency arrays. The averaged equations allow us to examine experimental strategies for obtaining higher power outputs from arrays. These strategies rely on resonant architectures that place the junctions near antinodes of a desired standing wave mode of the fluctuating current. Simple, heuristic rules are derived for the proper placement of junctions. The second part of the dissertation is devoted to stochastic resonance. A new theory is proposed to explain both two-state and excitable stochastic resonance. Previous theories explaining the two types of stochastic resonance yield similar results while using different analytic strategies. A constrained asymmetric rate model is derived that in one limit produces the proper result for the two-state system, while in another limit models the excitable system. The result that the constrained asymmetric rate model gives in the excitable limit is off by a factor of two, and this discrepancy is examined. Furthermore, we study the consequences of adding a colored noise source to the classic two-state model of stochastic resonance. We will find that when both white and colored noise sources are present, stochastic resonance will occur as a function of colored noise strength only if the correlation time of the colored noise source is small enough. Two theories are proposed to explain this phenomenon and both are examined in detail.