Peter Lamont Miller - US grants

NSF Award Abstract - DMS-0103909
Mathematical Sciences: The Semiclassical Limit of the Focusing Nonlinear Schroedinger Equation

Abstract

0103909 Miller



This project addresses the behavior of solutions of the focusing nonlinear Schroedinger equation in the singular semiclassical limit, with particular attention paid to solutions that are tied to definite given initial data. Specific research goals include (i) generalizing a steepest-descents procedure for matrix Riemann-Hilbert problems to recover asymptotics of the initial-value problem for general real-analytic and oscillatory-analytic initial data, (ii) computing rigorous spectral asymptotics for the nonselfadjoint Zakharov-Shabat operator and taking estimates of the error into account in the inverse-scattering problem, (iii) studying sets of minimal weighted Green's capacity in the upper half-plane and relating them solidly to semiclassical asymptotics, and (iv) determining the sensitivity of the asymptotics to the presence of singularities in the data and also robustness to structural perturbations. The analysis will employ numerical methods, careful asymptotic spectral analysis of a family of nonselfadjoint differential operators, and potential-theoretic aspects of functional and complex analysis.

The focusing nonlinear Schroedinger equation is a ubiquitous model equation for the propagation of waves of many different kinds (water waves, light waves, etc.) in the simultaneous presence of nonlinear effects that can "self-amplify" the waves and "dispersion" which can pull the waves apart. In particular, it is a tested and accepted model for the transmission of lightwave pulses along certain types of glass optical fibers. This project will produce new understanding of this model equation relevant to situations where the coefficient of the dispersive term in the equation is relatively small, or alternatively, nonlinear processes dominate the evolution of broad disturbances for short times. "Dispersion-shifted" optical fibers currently being installed in many modern telecommunication systems provide an environment where the effects of dispersion and nonlinearity are present in precisely such a skewed proportion. The results of this project will be likely to influence the analysis and design of the next generation of high-speed optical telecommunications systems.

DESCRIPTION (provided by applicant): Despite ample research on the efficacy of pharmacotherapy in the treatment of alcohol abuse and dependence, adjunctive medications are used infrequently in clinical settings. Unfortunately, evidence-based strategies to increase adoption of pharmacotherapies by alcoholism treatment practitioners are lacking. The primary objective of this exploratory research is to develop a quality improvement intervention strategy for pharmacotherapy transfer (using naltrexone as the prototype) and to pilot test and refine this intervention in community-based treatment centers. A unique feature of the study is that a team of university researchers, alcoholism counselors and alcoholism program administrators will collaborate in the development and testing of the intervention. Initially, a theoretically sound, multi-component intervention will be developed based on known attitudinal and organizational barriers to the use of pharmacotherapy in clinical practice. Interventional components will include an educational workshop, on-site academic detailing visits, audiovisual and written training materials, and ongoing email and telephone consultation. The intervention will be pilot testing at three experimental and three control alcoholism treatment centers over a six-month time period. Pre-post data on knowledge, attitudes and practices regarding naltrexone will be analyzed. After a thorough evaluation of pilot test results, the intervention will be refined and manualized in preparation for a more extensive controlled outcome study in a future investigation. The long-range significance of this study is that it will provide the first evidence-based translational model to facilitate adoption and implementation of current and future pharmacological agents in the treatment of alcohol abuse and dependence.

Abstract

DMS-0354373 FRG: Collaborative Research in Semiclassical Asymptotic Questions in Integrable Nonlinear Wave Theory

Peter Miller, Kenneth McLaughlin and Jared Bronski

The goal of applied mathematics is the study of equations of scientific,
engineering or industrial interest in a mathematically rigorous way.
Understanding such equations often requires considering
a limit in which certain parameters approach zero. While
some of these problems behave in a predictable manner in the
limit, other interesting and important problems involve unstable
behavior that becomes less and less predictable the smaller the
parameter of interest becomes. One of the best illustrations of this
concept is the behavior of fluid flows when the viscous drag is small,
and the fluid behaves in a turbulent and chaotic way.
The primary model under study in this project is the nonlinear
Schrodinger equation, which is a fundamental model for the study of
pulses in optical fibers. The small parameter limit for this
model corresponds to the limit of ultra-short pulse propagation, which
is expected to find many applications to high-speed telecommunications.
This limit is known as the semiclassical limit.
This project constitutes an in-depth study of initial-value problems
for several partial differential equations (PDEs) in the semiclassical
limit. Existing formal theories fail because they lead to model problems that are
ill-posed and thus make no prediction at all for reasonable initial
conditions. The goal of this project is to develop asymptotic
theories that are not based on any particular ansatz and do not
require unphysical conditions on the initial data. Among the problems
under attack is the rigorous semiclassical analysis of the focusing
nonlinear Schroedinger equation for general data, a problem that is
generally considered to be one of the most important open problems in
the field of integrable systems. The specific aims of the project
include the development of new ansatz-free methods of asymptotic
analysis --- for spectral theory and for Riemann-Hilbert problems of
inverse-scattering theory --- that are insensitive to analyticity
properties of the initial data, and have a "nonlinear
Riemann-Lebesgue" character, directly exploiting cancellation due to
oscillations where analytic deformations are impossible. These
techniques, once developed, will also have important repercussions in
fields that are only tangentially related, for example, the theory of
orthogonal polynomials of large degree and the statistical analysis of
large random matrices.

The main goal of the proposed work is to develop predictive tools that
apply to extremely unstable systems when the initial conditions
are rough or noisy. An example of such a system is the one governing the
propagation of ultrashort data pulses in certain optical fibers, and
therefor any new insights into such systems will have repercussions
in the field of telecommunications. Some of the problems we propose to
study admit a detailed analysis because they are idealized and can in some
sense be solved exactly; however our work is also expected to lead to general
methods that are applicable to less idealized systems.
These techniques will also have important implications in mathematical
fields that are only tangentially related, for example, the theory of
orthogonal polynomials of large degree and the statistical analysis of
large random matrices.This project also has an important educational
aspect. We plan to involve both graduate students and postdoctoral researchers in
this work, and the advanced training of the next generation of researchers
is an important component of this proposal. We also plan an
interdisciplinary workshop to further disseminate the results of this work beyond the
boundaries of the mathematical community.

DESCRIPTION (provided by applicant): A prime reason that alcohol screening of patients in primary care has been recommended is that heavy alcohol use can worsen medical conditions such as hypertension and diabetes mellitus. Unfortunately, alcohol screening in medical settings is not routine and little research is available on effective methods of facilitating the adoption and implementation of such screening by primary care providers. This project examines the effectiveness of the Practice Partner Research Network's (PPRNet) "Accelerating Alcohol Screening --Translating Research into Practice (AA-TRIP)" model to improve the detection, brief intervention, treatment (including pharmacotherapy and medical management) of alcohol problems by primary care physicians. The primary objective of the project is to compare the AA-TRIP model (with 10 primary care practices throughout the United States) to a control condition (10 primary care practices) in increasing the use of the Updated 2007 NIAAA guidelines for alcohol screening, diagnosis, brief intervention, and pharmacotherapy treatment by primary care medical practices for adult patients. We hypothesize that the AA-TRIP intervention model will result in significantly more screening, intervention and treatment (including medical management and pharmacotherapy) than the control condition. In addition, we are investigating the sustained effects of AA-TRIP on NIAAA alcohol screening after training and support have been discontinued. Secondary aims of the project include the investigation of practice-level variables related to implementation and sustainability of AA-TRIP using both quantitative and qualitative research methodology. Finally, we are investigating the impact of alcohol screening and treatment on disease-specific patient outcomes (i.e., change in blood pressure in HTN patients receiving brief intervention/treatment;change in hemoglobin A1c (HgbA1c) in DM patients receiving brief intervention and or treatment). The outcome could provide the first implementation model for pharmacotherapy and medical management of alcohol use disorders in primary care. In addition, results showing a relationship between alcohol intervention and improvement of hypertension and diabetes could have far-reaching implications for routine alcohol screening and treatment of patients with these medical conditions. Finally, increased treatment of alcohol use disorders in primary care could bring needed help to the 80% of alcoholic individuals in the USA who are currently not receiving treatment. Public Health Relevance: The results of this study could provide evidence for an effective method of increasing the use of alcoholism medications and medical management in primary health care with hypertensive and diabetic patients who have alcohol use disorders. The study could also show that primary care treatment of alcohol dependence leads to better blood pressure and blood glucose control in some patients.

DESCRIPTION (provided by applicant): Heavy alcohol use and alcohol use disorders in dental patients are associated with increased risk of oral and pharyngeal cancer, excessive bleeding during dental procedures, loss of tooth structure, periodontal disease, salivary gland disease, caries (cavities), and adverse drug reactions to local anesthetics and analgesics. Oral and pharyngeal cancer kills ~8,000 Americans each year. Dental students receive little education on alcohol use and no evidence-based, standardized curriculum is available to dental schools to educate their students about alcohol and oral health. The primary objective of this R25 educational grant application is to develop and evaluate the first online curriculum for dental schools to teach dental students about (1) the association between heavy alcohol use and oral health problems including oral and pharyngeal cancer, (2) systemic and oral biological effects of heavy alcohol use, (3) practical guidelines for the dental treatment of heavy drinking patients, (4) methods of screening dental patients for alcohol use, and (5) methods of brief intervention and referral to help patients reduce their alcohol use. This project is unique in that it will develop and evaluate the first online alcohol educational program specifically designed for dental students that can be integrated into dental school curricula. Exposure to this online program will increase dental student knowledge and skills at identifying and counseling patients experiencing or at-risk for oral health complications including oral and pharyngeal cancer. PUBLIC HEALTH RELEVANCE: This study could provide the first evidence-based alcohol education curriculum for dental students that can be integrated into dental school curricula. The online modules developed in this project will increase dental student knowledge and skills at identifying and counseling heavy drinking patients who are at risk for oral cancer and other oral health conditions and complications. Such identification could help dentists prevent complications, lower risk factors, and reduce morbidity and mortality.

This project concerns the development of semiclassical asymptotic techniques for initial-value problems and initial/boundary-value problems of nonlinear wave propagation in the situation that the equation of interest is completely integrable, possessing a Lax pair representation and all of the corresponding mathematical framework. Specific problems to be addressed include (i) the weakly dispersive asymptotics of the intermediate long-wave equation and its degeneration to the Benjamin-Ono equation, (ii) mixed initial/boundary value problems for the semiclassical defocusing nonlinear Schroedinger equation on the half-line, (iii) initial-value problems for the semiclassically scaled 3-wave interaction equations, (iv) transsonic initial-value problems for the semiclassical modified nonlinear Schroedinger equation, and (v) large-degree asymptotics for rational solutions of Painleve equations. Solving these problems will require the development of new tools of asymptotic analysis for nonlocal and higher-dimensional Riemann-Hilbert analytic factorization problems.

This project involves models for the propagation of waves of various sorts. In particular, the models are capable of making predictions about the behavior of physical systems ranging in diversity from optical fibers carrying high-intensity pulses of light to the ocean, where contaminants can become trapped at the boundary between layers of fluids of differing density. Our aim is to provide practical methods for accurately approximating the solutions of these models so that their predictions can be compared with experiments and ultimately used in place of expensive or invasive experiments themselves. In particular, we note that in the wake of environmental disasters like the Deepwater Horizon oil leak of 2010, it becomes increasingly important to have accurate and easy-to-use methods for predicting the location of contaminants in the fluid column of the ocean.

Waves in nature (for example, water waves or electromagnetic waves) can be modeled by solutions of certain differential equations that incorporate various physical processes into a mathematical framework that admits further detailed study in principle. But even with such a wave equation in hand, there remains the difficult task of deducing important information from the model, information that is needed to solve important problems of engineering that motivate the study of waves in the first place. One common approach is to use computers to solve the equations (approximately). However, such an approach is limited in scope to very concrete simulations involving particular initial conditions, and it holds only in parameter ranges in which the numerical methods can be accurate. On the other hand, there are other parameter regimes that are quite common (for example, the situation that electromagnetic waves can be approximated by light rays) in which the computer-based approach becomes difficult, and therefore one would like to have an alternative method of analysis. This project is aimed at developing such alternative methods of asymptotic analysis for wave propagation problems that are nonlinear (so that large waves can be accurately modeled) but that nonetheless admit a kind of transform relating them to linear problems (for which the familiar superposition principle applies). One application of such theoretical analysis would be to describe the evolution of sub-surface oil plumes caused by an oil leak like the Deepwater Horizon disaster.

This project is an attempt to place completely integrable nonlinear wave equations on a similar footing as constant-coefficient linear equations, from the point of view of asymptotic analysis (i.e., to further develop nonlinear analogues of the classical methods of stationary phase and steepest descent for integrals). The specific problems to be addressed include the study of the inverse-scattering transform for the Benjamin-Ono equation (a nonlocal integrable model for internal gravity waves) in the small-dispersion limit, the study of the resonant interaction of wave packets through a quadratic nonlinearity in the semiclassical limit, the study of an analogue of the defocusing nonlinear Schroedinger equation describing waves in two spatial dimensions in the semiclassical limit, the study of dynamical stability of so-called rogue waves, the study of mixed initial/boundary-value problems for integrable equations in the semiclassical limit, and the investigation of "universal wave patterns", analogues to wave propagation problems of universal phase transitions in statistical mechanics and mathematical physics.