Harvey Segur - US grants

Proposal # DMS-9731097 PI: Harvey Segur Title: Nonlinear Wave Motion The Kadomtsev-Petviashvili equation is a nonlinear partial differential equation in two spatial dimensions plus time. It is one of very few nonlinear equations that: (i) is completely integrable and admits "soliton" solutions; (ii) involves two spatial dimensions plus time; and (iii) arises naturally in physics. Specifically, the equation describes approximately the motion of ocean waves in shallow water. (Ocean waves require two spatial dimensions because they occur on the water's two-dimensional surface.) For this application, the most interesting solutions are either exactly or approximately periodic, just as typical ocean waves are approximately periodic. Based on recent discoveries of the mathematical structure of this equation, we can now solve this problem as an initial-value problem for initial data that are either exactly or approximately periodic in space. The work to be carried out in this study will exploit this structure to build an accurate model of waves in shallow water. The waves in the model will be as complicated as waves in the ocean usually are: steady or unsteady, with either exact or approximate periodicity in time or in space or both, with arbitrarily large amplitudes, and with fully two-dimensional spatial patterns. No other functioning model of water waves includes all of these features simultaneously, so this model should vastly improve our ability to predict waves in shallow water. Nonlinear phenomena occur when a perturbation to a system triggers a response that is not proportional to the perturbation. Many of the dramatic events in nature are nonlinear, including the "big bang" in cosmology, sonic booms in aerodynamics, hurricanes and breaking ocean waves in geophysics. The mathematical models used to describe these phenomena are intrinsically nonlinear, and typically we cannot solve them in any general sense. The set of models that admit "solitons" and are "completely integrable" are exceptions to this rule: usually these models can be solved in complete detail, and they have provided great insight into the nature of nonlinear wave phenomena. The work to be carried out in this study will exploit recently a discovered mathematical structure of one of these completely integrable models to build an accurate and realistic model of ocean waves in shallow water. Practical implications of more accurate predictions of shallow water waves include better control of beach erosion, fewer problems with shoreline pollution from dispersal of man-made waste, and better design of offshore structures like oil platforms.

Theoretical predictions are to be made for the characteristics of finite amplitude, shallow water waves without the restriction on one-dimensionality. Two independent spatial periods in the two horizontal directions will be modeled, representing long-crested and short-crested waves. The complete model will be enclosed in a simple software package.

This is a grant under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences of the National Sciences Foundation. This program supports the purchase of special purpose computing equipment dedicated to the conduct of research in the mathematical sciences. This equipment is required for several research projects and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers and with substantial cost-sharing from the institutions submitting the proposal. This program is an example of academic, corporate, and government cooperation in the support of the basic research in the mathematical sciences. This equipment will be used to support five research projects in the Program in Applied Mathematics of the University of Colorado, Boulder: aspects of solutions, integrable systems, and computation, directed by Mark Ablowitz; iterative methods for the solution of nonlinear equations, directed by James Curry; discrete dynamical systems, directed by Robert Easton; dynamics of Hamitonian mappings, directed by James Meiss; the Kadomtsev- Petviashvili equation and water waves, directed by Harvey Segur.

This grant is for an international conference on asymptotic expansion and perturbation theory to be held in January 1991 at San Diego, CA. The title of the conference, Asymptotics Beyond All Orders, refers to a small but significant class of problems in which conventional asymptotic methods provide no information at any order of the expansion. Problems of this nature are unusual, but not impossible: examples are known in hydrodynamics, plasma physics, crystal growth, and quantum mechanics. Several of these problems in widely different fields have been solved recently, using several apparently different methods. What the different methods of solution have in common is still largely unexplored. The purpose of this conference is to convene the principal participants in this nascent field for an in-depth comparison of their problems and their methods of solution.

This research encompasses three separate lines of study: a) Asymptotics beyond all orders; b) Kinetic theory and resonant triads, and c) Models for motion of curves and surfaces in space. The results have potential for important applications in nonlinear dynamics and applications of the KAM theory. The KAM theorem asserts that under appropriate circumstances most of the trajectories of the system remain regular at the onset of chaos; only a few trajectories actually become irregular, or chaotic. So far there is no available theory that can identify which trajectories become chaotic under small perturbations. If successful, the work proposed here will enable scientists to work with dynamical systems containing both chaotic and regular trajectories. This issue is important in many fields, one of which is the design of plasma fusion devices like tokomaks.

9422287 Segur Applicants to the NSF 1994 Minority Graduate Fellowship competition who were awarded "Honorable Mention" status and who enrolled in a computer science or computer engineering graduate program at a U.S. university were eligible to apply to the CISE Directorate for this special award. The purpose of the award is to assist the student in both research and educational activities related to his/her graduate education. The award is made on behalf of the student to the institution with the student's advisor designated as principal investigator. ***

9810751 Meiss The Department of Applied Mathematics of the University of Colorado at Boulder's VIGRE program is focussed in the areas of computational and nonlinear mathematics. The program creates tetrahedral research groups consisting of a faculty mentor, a postdoctoral fellow, several graduate students, and undergraduate majors. Tetrahedra will focus on particular interdisciplinary research problems, hold seminars, write research proposals, and participate in the curriculum reform program. The curriculum will be integrated with computation using multi-layered case study modules. Affiliated faculty serve as co-advisors for Ph.D. theses of our students; the affiliated faculty program will be extended to government laboratories and technology companies. The program will create four tetrahedral groups in the areas of Dynamical Systems, Nonlinear Waves, Multilevel Computation, and Fast Algorithms and Modeling. The four facets of interaction within the groups include: teaching--as a seminar, and in the development of case study projects; learning--to develop mathematical, computational, and communication skills; discovering--to develop the techniques for formulation of useful and solvable research problems; and communicating--to collaborate in the joint production of research papers, grant proposals, and interim research reports. Computational mathematics provides the unifying theme for vertical integration of our training program. Computation will be integrated into lower division courses through Case Study Modules. Vertical integration will be implemented through the development of a multi-layered modeling course that has both lower and upper division undergraduate components as well as a graduate component. The VIGRE grant will support four Postdoctoral Fellows. They will receive mentoring from a faculty advisor and teach one course each semester for the Department. Twelve Graduate Trainees will be funded by the proposal. They will participate in the tetrahedra, research pro posal development, and the development and implementation of case study modules and receive teacher training through the Teaching and Learning Seminar and the Graduate Teacher Program. Four Undergraduate Research Experiences will be funded each year. Students will participate in one of the tetrahedra part-time during the academic year and for two months during the summer. These initiatives will be sustainable and lead to a number of permanent structural changes: successful tetrahedral research groups will endure and be emulated, case study modules will form an essential part of our curriculum, and the extended affiliated faculty will result in a continuing option for training of our students in application areas. Funding for this activity will be provided by the Division of Mathematical Sciences and the MPS Office of Multidisciplinary Activities.

The overall objectives of this work are to develop a thorough
understanding of three-dimensional water waves of finite amplitude, and
ultimately to develop a practical model to describe these waves efficiently.
A model that is both accurate and computationally efficient could have
many practical applications. Specific problems to be addressed are: (1) the
existence and stability of three-dimensional, doubly-periodic, traveling
water-wave patterns, through the full range of depths; (2) the prevalence
of hexagonal, rectangular or crescent-shaped waves (or other multiply
periodic wave patterns) among ocean waves; (3) the long-wave and
modulational descriptions of water waves, and the subsequent stability
analyses that are feasible in these cases; (4) the design and implementation
of algorithms to make practical use of exact solutions of asymptotic
models in shallow and deep water; (5) the relation between the detailed
dynamics of three-dimensional, nonlinear waves and some commonly
used ocean-wave transport models; and (6) the impact of a detailed local
description of nonlinear wave dynamics on these transport models, in the
presence of large amplitude nonlinear waves or under conditions of
nonlinear wave focusing. These problems will be studied using analysis,
computation, asymptotics, and algebraic geometry, involving the full
equations and approximate models, all in conjunction with state-of-the-art
physical experiments.

The destructive force of large-amplitude ocean waves is well known.
Large-scale ocean waves have a major impact on the design of ocean-
going ships, of off-shore oil platforms, and of other structures in a coastal
environment. These waves also impact the scheduling and routing of
shipping patterns, and they strongly affect air-sea transport processes. Yet
most theoretical models of ocean waves now in use are based on waves of
small amplitude. In this investigation we focus on developing a thorough
understanding of large-amplitude waves. The ultimate goal is to develop a
practical, mathematical model that may be used operationally in the
applications listed above. In particular, the investigators plan to build on
their recent work in which they have observed certain coherent patterns of
large-amplitude waves. They have observed these patterns in laboratory
experiments, as solutions to the well-known equations of water waves, and
as solutions to other equations that are (more) approximate models of
water waves. Their work involves a variety of mathematical and
computational tools as well as state-of-the-art laboratory experiments. In
the present work the investigators will combine all of their tools to
understand and describe these coherent patterns and to use them as the
building blocks for a practical model of ocean waves.

The Departments of Mathematics and Applied Mathematics at
the University of Colorado will conduct research efforts
that will purchase two "compute engines", one for each of
two departments, that incorporate these advances. Among
advances in computing capacity two are crucial for the
problems of interest here: (i) increased size of memory now
allows for realistic computations of some physical phenomena
that evolve in time in three spatial dimensions;
and (ii) increased computational precision makes feasible a
new class of fast algorithms that were infeasible otherwise.

1) A goal of one project is the development of fast and
accurate algorithms for solving practical problems in
dimensions three and higher. The two main tools in this
development are representations with a low separation rank
(a numerical version of separation of variables).

2) A goal of a second project is prototyping new algorithms
and developing new models for moderate to high Reynolds
number flows.

3) A third research project, on the behavior of a granular
medium in two and three dimensions, draws on the classical
kinetic theory, extending it to the case of inelastic
collisions, and gives explicit expression for macroscopic
quantities such as stress and granular energy.

Scientific problems of immediate interest include:
(a) geophysical and astrophysical fluid flows, driven by
the competing influences of planetary rotation, variable
gravity, and magnetic forces; (b) granular flows, such as
occur in avalanches or sand-piles or landslides; and (c)
electromagnetic wave propagation near irregularly shaped
bodies. Aside from these immediate applications, a
longer-term goal is to train students to use these new
machines well. The University of Colorado recently
created two new graduate programs, designed to train
scientists and engineers in advanced computational methods.
These two machines will help to make these new programs
more effective.

The MCTP: Colorado Advantage proposal will build on existing programs and efforts within the department as well as on campus (e.g. SMART program). MCTP: Colorado Advantage will continue to attract numbers of underrepresented students to the mathematical sciences by actively collaborating with the Colorado Diversity Initiative. Support from NSF-funded MCTP: Colorado Advantage will allow the department to develop and implement strategies for sustainability, such as time for re-tasking existing department resources, working with the Deans (Engineering and Arts & Sciences Deans) and Provost to gain additional resources as well as working with successful donors capable of endowing undergraduate scholarships. The 17 faculty members in the department strongly endorse this MCTP proposal. The intellectual merit of this proposal is to introduce a large number of undergraduates to the excitement of research and to stimulate their interest in furthering their mathematical education. The ability of the Department to meet this objective can be inferred from its record with its previous VIGRE grant, which began in 1999, and trained 54 undergraduates.

Among the broader impacts of the MCTP: Colorado Advantage program are that it will significantly increase the number of students who take more advanced mathematics courses; the number of majors who have the transformative opportunity of working on serious longer term projects and research projects; and the number of students who graduate from the Department of Applied Mathematics and are well prepared for graduate school and the scientific workforce. Further its multiplier effect on other undergraduates at the University through the creation of a culture of undergraduate research activity will have longer-term implications for the University as a whole. An additional impact will be on graduate students and faculty who become involved in these critical mentorship activities as they pursue their own careers. Finally, the MCTP: Colorado Advantage proposal is a model that can be used at any research university that has a graduate program, faculty committed to undergraduate education and a will to transform itself.

Many physical systems admit nonlinear, dispersive waves, for which the speed of propagation depends on both the frequency and amplitude of the wave. These systems are usually dissipative, but the dissipation might be weak. This proposal focusses on nonlinear dispersive waves in the presence of weak dissipation, with special emphasis on surface water waves. Recent work by the PIs of this proposal and others has shown that even weak dissipation can profoundly affect the stability of nonlinear dispersive waves in certain situations. The overall theme of this proposal is to study nonlinear dispersive waves in the presence of weak dissipation. We will use a state-of-the-art laboratory facility to conduct experiments on waves on deep water. These experiments will guide the development of mathematical models and test their validity. Questions we will explore include (a) How does weak dissipation affect the stability and properties of surface wave patterns that are more complicated than ordinary plane waves? (b) What causes the downshifting of the peak of a narrow-banded spectrum of nonlinear, dispersive waves, as observed in experiments on water waves and in optics? (c) Does this theoretical and laboratory work apply to actual ocean waves? (d) What causes the dissipation in surface water waves?

Particular goals of this research include obtaining better predictive models of the dynamics of large-amplitude ocean waves, and of exchange processes that occur at the ocean-atmosphere boundary. Accurate models of these exchange processes will be essential in predicting global warming and related global phenomena. In addition, accurate models of wave dynamics might prove useful in understanding extreme waves such as rogue waves. Much of the work to be conducted is fundamental in nature and the mathematics is shared by many physical systems, so the results obtained for water waves could affect our understanding of many physical systems. These other systems include light waves in an optical fiber, spin waves in a thin magnetic film and others.

Mathematics (21) This collaborative project addresses a problem that occurs in many mathematics and mathematics-related courses at the university-level across the US: high failure rates in important early college mathematics courses, especially in Calculus I. The teaching strategy being used and tested is based on the idea of using Enhanced Conceptual Development through Focused Oral Discourse, or Orals. The current work is analyzing and extending earlier work that has been focused on helping students identified as at risk of failing calculus. Based on the earlier success of reducing failure rates using effective teaching strategies, including Orals, the project team is now applying these teaching strategies to diverse users in several new settings: to classes taught in different STEM (science, technology engineering, and mathematics) departments, at a different college, and at different educational levels.

The goals of this project are to: 1) Refine, implement, and test Orals with diverse groups of learners and in diverse education settings; 2) Provide training, coaching and evaluation for facilitators of Orals including Instructors, Teaching Assistants and Undergraduate Learning Assistants; 3) Provide extensive assessment artifacts of the implementations; 4) Create a database and website of new learning materials (Orals questions for diverse courses); and 5) Improve the retention and understanding of STEM students.

Intellectual Merit: The importance of discourse in the mathematical sciences classroom has already been shown for K-12. A potentially important impact of this proposal will be to contribute to the national dialogue by conducting careful experiments that assess one method of increasing discourse: Orals. In particular, the project team will conduct proof of concept studies of Orals, in moving from small classrooms (the original setting) to large lecture sections.

Broader Impact: Many STEM majors require successful completion of a calculus sequence; however, many university students do not achieve their career goals because of their inability to pass the introductory calculus courses. This project will address this important national issue. The new work will not only be in a larger variety of mathematics courses, but also in introductory Aerospace and Mechanical Engineering courses, and at a local high school in a two-year algebra course.

Many physical systems admit nonlinear waves. These systems often exhibit dissipation that is considered to be negligible or weak. However, recent work by the investigators and recently published observations of dissipation rates of ocean swell, show that the models using weak dissipation provide an inadequate description of some aspects of the evolution of nonlinear surface water waves. Thus, a main emphasis of this research is to develop a theory from first principles for the propagation and evolution of nonlinear surface water waves on a viscous fluid, with no approximation on the size of dissipation. The research addresses several projects that build on this notion and/or are intended to apply the mathematical results to ocean applications. The projects fall into two categories: A) surface waves propagating on a viscous fluid and B) waves in shallow or finite-depth water. The investigators will approach these problems by combining their individual strengths in modeling and analysis, numerical simulations, and physical experiments to obtain a fundamental mathematical description of the physical phenomena. They will then use publicly available data from ocean observations to apply their results to ocean settings.

The primary goal of this research is to understand at a fundamental, mathematical level the propagation of waves on a surface of water, and to apply this understanding to observations of waves measured in the laboratory and in the oceans. Particular applications include the effects of dissipation on the generation of waves by wind, the subsequent propagation of ocean swell, and the development of rogue waves in shallow and deep water; predicting dangerous versus benign tsunamis given the Richter-scale measure of the magnitude of the responsible earthquake and some readily available information about the geology of its location; and modeling large-amplitude waves on beaches with variable bathymetry using the new mathematical approaches investigated here. In addition, the mathematical results are expected to describe nonlinear waves in several settings, including light waves in an optical fiber, spin waves in a thin magnetic film and others. Graduate and undergraduate students will be mentored and involved in experimental studies and theoretical analysis; through this project they will receive training in interdisciplinary research.

Waves on the ocean's surface play important roles in weather forecasting and climate modeling, in the safety of coastal communities and offshore industries, and in overseas shipping. In this project, the investigators focus on physical effects that are often approximated or neglected altogether, but that are needed to predict accurately the observed behavior of ocean waves. Examples include the dissipation of ocean swell during propagation across the deep ocean and subsequently onto the shoreline; the time-dependence of wind in the wave-generation process; and dispersion of waves in shallow water. The inclusion of dissipation will lead to a better understanding of how wave energy evolves. The inclusion of time-dependence in the wind that generates waves will allow for a better understanding of the initial period during which energy is transferred from air to water. The inclusion of dissipation and dispersion in models for shallow-water waves will allow for better predictive capabilities of waves in coastal areas. The research tools of the investigators include modeling, analysis, computer simulations, and laboratory experiments. While the emphasis is on water waves, for which they can conduct laboratory experiments, the mathematical analysis is more broadly applicable to other physical systems and is of interest in the study of partial differential equations.

The investigators propose analytic, numerical, and experimental investigations of the following: (A) Deep-water waves. They consider the frequency downshifting of freely propagating waves and wave generation due to wind. They are considering two models of frequency downshifting that differ in how the rotational part of the flow is modeled. To model wind-generated waves, they are allowing for time-dependent shear flows in both the air and water. The resulting stability problem for waves is non-standard, and understanding how to address it is a central mathematical question. (B) Shallow-water waves. They seek accurate models of dispersion and of dissipation due to the bottom, wall, and surface boundary layers. They will start with a Whitham equation and generalize it to include surface tension effects, dissipative effects, nonhorizontal bathymetry, and bidirectional waves. They will look, both analytically and numerically, for small- and large-amplitude solutions, and study their stability. They will further investigate how best to include dissipation that is due to the bottom boundary layer by comparing numerical simulations and experiments. (C) Three-wave partial differential equations. The three-wave partial differential equations, which arise in many physical applications, describe the simplest possible nonlinear interactions among dispersive wave trains, without dissipation. The investigators propose a solution method using a Painleve-analysis to obtain the general solution for arbitrary boundary conditions. There are few examples of general solutions of partial differential equations, so their adding one more example would be a mathematical breakthrough.