Peter 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.

High-speed fiber optics plays a key enabling role in today's information technology revolution by providing a means for high bandwidth communications. Commercial optical transmission bandwidths are increasing at an impressive rate. In light of these advances, new effects which were previously considered insignificant are now viewed as key limiting factors in high performance lightwave systems. In order to sustain the growth in bandwidth, new technologies are needed to compensate new transmission impairments that arise with rapidly growing data rates. One such impairment, which has become a key factor limiting transmission rates and distances in high-speed fiber systems, is an effect called polarization-mode dispersion (PMD). PMD arises due to small, spatially varying random birefringences in optical fibers, which lead to decorrelation of the input and output polarization states and pulse spreading in the time domain. The latter effect can cause errors in digital communication systems. This problem is especially acute in the embedded fiber base where many fibers exhibit strong PMD.

This proposal describes a university-industry collaborative project which aims to demonstrate a novel and advantageous method for compensation of PMD. The concept is to exploit and extend technology developed in the field of ultrafast optical pulse shaping to compensate PMD of wide-band optical signals in parallel on a wavelength-by-wavelength basis and under computer control. This improves on current approaches that only allow compensation of so-called first order PMD for a single wavelength and which do not apply to situations with wide-band optical signals where the distortion caused by PMD varies substantially across the optical bandwidth.

This research project will encompass several aspects. One goal is to perform experiments demonstrating the validity of the proposed new PMD compensation scheme. Research will also target programmable generation of PMD for use in test equipment. Both of these activities will be backed up by numerical simulations. Additionally, new PMD measurement strategies will be devised in order to provide the data necessary to suitably control the proposed wavelength-by-wavelength PMD compensators. This should also result in a major advance in instrumentation for state-of-polarization sensing applicable to multiple-wavelength 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.

DESCRIPTION (provided by applicant): Hypertension affects 50 million adults and is the leading cause of stroke and congestive heart failure in the United States. While many factors contribute to hypertension, there is overwhelming evidence linking excessive alcohol consumption to increased blood pressure. Either total abstinence or reduction in consumption to one drink a day results in a rapid and significant drop in blood pressure in many patients. Unfortunately, routine alcohol screening and intervention with hypertensive patients is rare in spite of the availability of clinical guidelines and screening tools. The primary objective of this study is to utilize the Practice Partner Research Network's Translating Research into Practice (PPRNet-TRIP) model to improve detection and management of excessive drinking among primary care patients with hypertension. PPRNet-TRIP is a validated, practice-based quality improvement system using electronic medical records, reminder prompts, academic detailing and performance feedback. An enhanced PPRNet-TRIP model, providing concentrated focus on alcohol screening for hypertensive patients, will be applied to ten primary care practices to improve detection and management of alcohol problems among patients with hypertension. Ten control practices will receive a more general quality improvement program without special emphasis on alcohol screening. A secondary goal of the study is to evaluate whether the enhanced intervention has a greater impact on reductions in blood pressure in hypertensive patients than the control condition. The significance of this study is that it may provide an evidence-based educational intervention to facilitate the routine use of alcohol screening with patients whose hypertension may be exacerbated by excessive alcohol consumption. Such screening and intervention should improve blood pressure control in many of these patients and, in turn, reduce risk of chronic disease and death.

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.

High-speed fiber optics plays a key role enabling high bandwidth communications. In light of impressive commercial advances, new effects have become key limiting factors for continued growth in the bandwidth of lightwave systems. One such impairment, viewed as a key roadblock for future high-speed systems, is an effect called polarization-mode dispersion (PMD). PMD arises due to small random variations along the lengths of optical fibers, which lead to scrambling of polarization states and pulse spreading in the time domain. The latter effect can cause errors in digital communication systems. This problem is especially acute in the embedded fiber base where many fibers exhibit strong PMD. This proposal describes a university-industry collaborative project which aims to demonstrate a novel and advantageous method for compensation of PMD. The concept is to exploit and extend technology developed in the field of ultrafast optics for wavelength-parallel sensing and compensation of wide-band optical signals. This improves on current approaches that only allow compensation of so-called first order PMD for a single wavelength and which do not apply to situations with wide-band optical signals where the distortion caused by PMD varies substantially across the optical bandwidth.
This research has the potential for broad societal impact by supporting advances in the high-speed communications infrastructure that has become pervasive to our economy. The participation of two industrial organizations as co-PIs in this proposal should catalyze rapid transfer of research results to practice. This research project should furnish excellent opportunities for broad student training in areas of cutting-edge technology, while providing teaming opportunities, including industrial research visits, that will enrich the students' educational experience.

This project concerns the development of tools for the asymptotic analysis of integrable nonlinear wave equations. There are two complementary aspects of this work: asymptotic analysis in the scattering theory of linear differential equations, and asymptotic analysis of the corresponding inverse problems. The particular application problems to be studied along the way include (i) the semiclassical limit of the focusing nonlinear Schrodinger equation, (ii) the corresponding semiclassical limit of the modified nonlinear Schrodinger equation, (iii) the semiclassical limit of the sine-Gordon equation in laboratory coordinates, (iv) the continuum limit of the Ablowitz-Ladik equations, (v) singular limits for multicomponent integrable equations, and (vi) several asymptotic problems in approximation theory with applications to random matrix theory.

This work is interesting and important because it will promote understanding of singular limits leading formally to ill-posed dynamical systems. Indeed, the motivating problem of the semiclassical limit of the focusing nonlinear Schrodinger equation with general smooth but nonanalytic initial data remains one of the most important open problems in applied analysis, and the tools developed as part of this project will directly address this problem and other similar ones. Furthermore, while the basic aim of the project is the development of methods of analysis, the methods will also be applied to several specific integrable equations and also to open problems beyond the field of nonlinear waves. For example, we intend to apply specialized asymptotic methods developed in the context of nonlinear wave theory to the problem of the asymptotic analysis (in the bulk scaling limit) of the correlation functions of the normal random matrix model, with its coincident connections to the theory of Laplacian growth (also known as Hele-Shaw flow) and conformal mapping. Many of the problems addressed as part of this project have a universal character, arising in the modeling of diverse physical phenomena, and it follows that analytical techniques applicable to these problems have far-reaching consequences. This project also has an educational component, stressing the training of postdocs and graduate students through collaborative research and course development.

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 study seeks to develop recall aids for the name generator procedure from the General Social Survey and examine empirically whether these aids can improve the recall accuracy of the information about who comprises their personal networks from survey participants. It hypothesizes that researchers can obtain more comprehensive personal network data by encouraging survey respondents to consult the actual records that they keep in the contact directories provided by various ICTs (such as the phone book stored in a mobile phone and the address book functionality of email applications). Thus far, although the past literature has suggested a few techniques to reduce respondents? burden in the survey setting, there is little work addressing the issue of the recall accuracy for personal network data collection.

This study employs a survey experiment; a Web survey will be administered to college students to gather information about their personal networks. College students consist of a homogeneous sample appropriate for this study, given the concern of internal validity in the expected findings. Students who agree to participate in the survey will be randomly assigned to three conditions. The control group will take the questionnaire without any recall aid, while the two experimental groups will take the survey with two different forms of prompts and probes respectively.

Broader impacts: The study will provide an effective technique to collect personal network information from individuals such as scientists and engineers. The proposed technique can then be used in surveys to collect information to develop new social capital indicators for the science and engineering workforce. As a result, researchers can use these indicators to investigate how various dimensions (e.g. advice, support, etc.) of personal networks may explain the productivity and career outcomes of scientists and engineers. More generally, this project will advance the understanding about individuals' personal networks as well as the data collection technique for personal network research. It will also offer new insights into the understanding of the psychology of survey response.

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.

The meeting "Scattering and Inverse-Scattering in Multi-Dimensions" will be held on the campus of the University of Kentucky, Lexington, KY, during the period May 15-23, 2014. The meeting features three days of conference talks (including tutorials) on May 15-17 followed by five days of workshop activity on May 19-23, separated by a break on May 18. The purpose of this conference is to bring together workers in integrable systems, dispersive nonlinear equations, and inverse scattering to forge collaborative links and advance research on completely integrable, two-dimensional systems, particularly dispersive models of nonlinear waves and normal matrix models. The meeting will pursue several lines of inquiry: (1) semiclassical limits of the direct and inverse scattering transforms, (2) eigenvalue distributions of random normal matrices, (3) exceptional sets and soliton solutions, and (4) one-dimensional limits of two-dimensional integrable systems.

This conference will bring together participants with expertise in pure and applied mathematics to attack problems with relevance for applications in physics, engineering, medicine, and oceanography. Applications to physics include models of random media, solvable models in statistical physics, mesoscopic physics, biophysical models, and water wave theory. Applications to engineering and medicine include non-destructive testing of materials and medical imaging. A common mathematical structure underlies all of these applications: the goal is to make progress in the understanding of applications in these diverse areas through collaboration among pure mathematicians, applied mathematicians, and numerical analysts. The award supports participation in the meeting by twenty junior researchers and fifteen senior researchers.

Meeting web site: math.as.uky.edu/scattering-and-inverse-scattering-multidimensions

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.

This project studies the phenomenon of universality in the context of models for the motion of large waves in several physical settings such as surface water waves and electromagnetic waves in optical fibers. Universality refers to situations in which the same or very similar wave patterns appear despite the waves being set into motion by quite different mechanisms, or even in quite different physical systems. For instance, rogue waves on the sea surface are frequently characterized as consisting of a large central peak with a distinctive dip on either side. It does not matter much the conditions under which the rogue wave appears — the pattern is nearly always the same. The aim of this research is to determine what patterns should be expected, as this knowledge can be used in applications ranging from device design to disaster mitigation, and why they occur. Furthermore, because modeling is a process that involves numerous ad-hoc assumptions, it is important to understand which features predicted by a model are independent of those assumptions, and universality gets to the heart of this question. Graduate students and early-career researchers will join the investigator in this study, which enhances its impact beyond scientific inquiry and into education and training of the next generation of scientists. <br/><br/>Mathematically, the study of universality is related to asymptotic analysis, specifically involving double-scaling limits to localize the coordinates near a point of interest, while a parameter in the model or solution becomes large. The investigator will study such double-scaling limits in various asymptotic models for nonlinear waves given by integrable evolution equations. Broadening the scope slightly, several specific questions involving asymptotic analysis of mathematical models for nonlinear waves will be addressed, including (i) determining the small-dispersion asymptotics of solutions of the defocusing nonlinear Schrödinger equation and the intermediate long-wave and Benjamin-Ono equations; (ii) analyzing the features of a new family of transcendental solutions of the focusing nonlinear Schrödinger equation termed "rogue waves of infinite order"; (iii) studying the degeneration of specific solution families of Painlevé equations. The investigator will combine and further develop techniques from the fields of integrable systems and asymptotic analysis to address these questions. Anticipated outcomes include a first proof of universal wave breaking in the defocusing nonlinear Schrödinger equation, new results on the small-dispersion asymptotic behavior of solutions of the intermediate long-wave equation (a nonlocal model for internal waves in stratified fluids interpolating between the shallow-water Korteweg-de Vries limit and the deep-water Benjamin-Ono limit), development of a new analytical technique for asymptotic analysis of nonlocal Riemann-Hilbert problems, and the discovery of new information about the solution space of Painlevé equations and the focusing nonlinear Schrödinger equation.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.