A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schr\"odinger Equations

A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schr\" odinger type equations. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlinearities that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions.Comment: LaTeX, 16 page

Using Stories in Coach Education

The purpose of this paper is to illustrate how storied representations of research can be used as an effective pedagogical tool in coach education. During a series of continuing professional development seminars for professional golf coaches, we presented our research in the form of stories and poems which were created in an effort to evoke and communicate the lived experiences of elite professional golfers. Following these presentations, we obtained written responses to the stories from 53 experienced coaches who attended the seminars. Analysis of this data revealed three ways in which coaches responded to the stories: (i) questioning; (ii) summarising; and (iii) incorporating. We conclude that these responses illustrate the potential of storied forms of representation to enhance professional development through stimulating reflective practice and increasing understanding of holistic, person-centred approaches to coaching athletes in high-performance sport

Existence and stability of hole solutions to complex Ginzburg-Landau equations

We consider the existence and stability of the hole, or dark soliton, solution to a Ginzburg-Landau perturbation of the defocusing nonlinear Schroedinger equation (NLS), and to the nearly real complex Ginzburg-Landau equation (CGL). By using dynamical systems techniques, it is shown that the dark soliton can persist as either a regular perturbation or a singular perturbation of that which exists for the NLS. When considering the stability of the soliton, a major difficulty which must be overcome is that eigenvalues may bifurcate out of the continuous spectrum, i.e., an edge bifurcation may occur. Since the continuous spectrum for the NLS covers the imaginary axis, and since for the CGL it touches the origin, such a bifurcation may lead to an unstable wave. An additional important consideration is that an edge bifurcation can happen even if there are no eigenvalues embedded in the continuous spectrum. Building on and refining ideas first presented in Kapitula and Sandstede (Physica D, 1998) and Kapitula (SIAM J. Math. Anal., 1999), we show that when the wave persists as a regular perturbation, at most three eigenvalues will bifurcate out of the continuous spectrum. Furthermore, we precisely track these bifurcating eigenvalues, and thus are able to give conditions for which the perturbed wave will be stable. For the NLS the results are an improvement and refinement of previous work, while the results for the CGL are new. The techniques presented are very general and are therefore applicable to a much larger class of problems than those considered here.Comment: 41 pages, 4 figures, submitte

Nonlinear ac response of anisotropic composites

When a suspension consisting of dielectric particles having nonlinear characteristics is subjected to a sinusoidal (ac) field, the electrical response will in general consist of ac fields at frequencies of the higher-order harmonics. These ac responses will also be anisotropic. In this work, a self-consistent formalism has been employed to compute the induced dipole moment for suspensions in which the suspended particles have nonlinear characteristics, in an attempt to investigate the anisotropy in the ac response. The results showed that the harmonics of the induced dipole moment and the local electric field are both increased as the anisotropy increases for the longitudinal field case, while the harmonics are decreased as the anisotropy increases for the transverse field case. These results are qualitatively understood with the spectral representation. Thus, by measuring the ac responses both parallel and perpendicular to the uniaxial anisotropic axis of the field-induced structures, it is possible to perform a real-time monitoring of the field-induced aggregation process.Comment: 14 pages and 4 eps figure

Geothermal probabilistic cost study

A tool is presented to quantify the risks of geothermal projects, the Geothermal Probabilistic Cost Model (GPCM). The GPCM model was used to evaluate a geothermal reservoir for a binary-cycle electric plant at Heber, California. Three institutional aspects of the geothermal risk which can shift the risk among different agents was analyzed. The leasing of geothermal land, contracting between the producer and the user of the geothermal heat, and insurance against faulty performance were examined

A Bayesian approach to Lagrangian data assimilation

Lagrangian data arise from instruments that are carried by the flow in a fluid field. Assimilation of such data into ocean models presents a challenge due to the potential complexity of Lagrangian trajectories in relatively simple flow fields. We adopt a Bayesian perspective on this problem and thereby take account of the fully non-linear features of the underlying model. In the perfect model scenario, the posterior distribution for the initial state of the system contains all the information that can be extracted from a given realization of observations and the model dynamics. We work in the smoothing context in which the posterior on the initial conditions is determined by future observations. This posterior distribution gives the optimal ensemble to be used in data assimilation. The issue then is sampling this distribution. We develop, implement, and test sampling methods, based on Markov-chain Monte Carlo (MCMC), which are particularly well suited to the low-dimensional, but highly non-linear, nature of Lagrangian data. We compare these methods to the well-established ensemble Kalman filter (EnKF) approach. It is seen that the MCMC based methods correctly sample the desired posterior distribution whereas the EnKF may fail due to infrequent observations or non-linear structures in the underlying flow

Existence and Stability of N-Pulses on Optical Fibers with Phase-Sensitive Amplifiers

The propagation of pulses in optical communication systems in which attenuation is compensated by phase-sensitive amplifiers is investigated. A central issue is whether optical fibers are capable of carrying several pieces of information at the same time. In this paper, multiple pulses are shown to exist for a fourth-order nonlinear diffusion model due to Kutz and co-workers [10]. Moreover, criteria are derived for determining which of these pulses are stable. The pulses arise in a reversible orbit-flip, a homoclinic bifurcation investigated here for the first time. Numerical simulations are used to study multiple pulses far away from the actual bifurcation point. They confirm that properties of the multiple pulses including their stability are surprisingly well predicted by the analysis carried out near the bifurcation