John Hunter - US grants

This research will examine high-frequency nonlinear waves with applications to problems in continuum mechanics. Asymptotic methodology will be developed to analyze hyperbolic waves in several space dimensions and resonant wave-wave interactions. Application areas include mach configurations in reflected, focusing, and refracted shocks, nonlinear caustics, resonant interaction of sawtoothed waves, passage through resonance of nonlinear waves, nondispersive surface waves, and weakly dispersive short waves.

This research addresses the analysis of nonlinear hyperbolic partial differential equations, with applications to wave propagation in a variety of physical contexts. The analysis is based on the systematic reduction of complex systems of equations to simpler model equations by means of asymptotic equations. The model equations are studied using a combination of explicit solutions, perturbation methods, qualitative analysis, and numerical calculations. The research lies in the general area of applied mathematics. Physical systems in which disturbances propagate at finite speeds are often modelled by hyperbolic partial differential equations. Examples include compressible gas flows, elasticity, magnetohydrodynamics, and general relativity. When the speed of the disturbance depends on the strength of the disturbance, the equations are nonlinear. An outstanding property of nonlinear hyperbolic waves is the formation of shocks. Shock waves are of significant practical importance (e.g. in sonic booms or the nonsurgical destruction of kidney stones) and pose many challenging mathematical problems.

9404152 Hunter The goal of the proposed work is to understand the behavior of nonlinear hyperbolic waves in continuum mechanics and classical field theory. The main approach is to use singular perturbation methods to derive reduced equations which capture essential nonlinear phenomena in their simplest form. These reduced equations are then studied using modern analytical and numerical methods. Specific problems include the transition from regular to irregular reflection in weak shock reflection, nonlinear waves in random media, and nonlinear waves in classical field theories. Many kinds of waves --- such as sound waves, elastic waves, and magnetohydrodynamic waves --- are described by hyperbolic partial differential equations. At large enough amplitudes, these waves are nonlinear. The effects of nonlinearity are often dramatic and are qualitatively different from the predictions of a linear theory. For example, nonlinearity leads to the formation of shocks, or sometimes to other kinds of singularities, and to various kinds of nonlinear instabilities. This proposal concerns a wide variety of physical problems involving nonlinear hyperbolic waves. The problems include the reflection of shock waves, the propagation of large amplitude sound waves through turbulent fluids, and the analysis of orientation waves in a massive liquid crystal director field.

9508411 Puckett The Department of Mathematics at U. C. Davis proposes to purchase a DEC Alpha 2100 computer and peripheral devices. This equipment will be used to support research in Large Scale Scientific Computation, Geometry, Asymptotic Analysis and Mathematical Physics. This research will include the use of symbolic mathematics computer languages and the development of graphical interface tools for interactive computing and the display of data.

9704155 Hunter The proposal is to study a number of nonlinear partial differential equations which model various physical and biological systems. The first equation is a free-boundary value problem which models the deposition of surfactant on a surface by a liquid droplet. There is a complex interaction between the motion of the droplet on the surface and the deposition of surfactant. The second equation is a system of reaction-diffusion equations which models an islet of beta-cells in the pancreas. The aim of Professor Hunter's research is to understand the collective dynamics of the beta-cells and the formation of spatial patterns in an islet. The third equation is a two dimensional Burgers equation which models the diffraction and reflection of weak shock waves in two space dimensions, where there are longstanding discrepancies between theory and experiment. The fourth equation is a modulation equation which describes the propagation of nonlinear gravitational waves in Einstein's theory of general relativity. Professor Hunter proposes to study this equation with the aim of understanding the formation of space-time singularities in a nonlinear gravitational wave. Mathematical models of physical and biological systems provide a way to understand, predict, and control the behavior of those systems. The proposed research involves the study of a number of mathematical models of systems of importance in engineering, biology, and basic science. One model describes the deposition of surfactant on a surface by droplets. Surface deposition has many applications in materials science, including the printing of very small structures on surfaces for the control of fluids, and as an experimental means of preparing liquid crystal display screens. A second model describes the collective behavior of beta-cells in the pancreas. These cells produce insulin and play a central role in diseases like diabetes. A third model describes the propagation of shock waves, which are generated by transonic or supersonic aircraft, in turbines, and in many other high-speed fluid flows. The mathematical theory of shock waves remains very poorly understood. The fourth model is a set of equations which describe the propagation of nonlinear gravitational waves in Einstein's general theory of relativity. The general theory of relativity is the fundamental physical theory of gravity, and it describes the large scale cosmological structure of the universe. The main obstacle to understanding the predictions of the theory is the nonlinearity of the Einstein equations. This work is directed towards an increased understanding of the effects of this nonlinearity.

NSF Award Abstract - DMS-0072343
Mathematical Sciences: Nonlinear Wave Propagation


Abstract

0072343 Hunter




This project studies a number of problems in nonlinear wave propagation. The first problem is the reflection of weak shock waves, where there have been longstanding discrepancies between theory and experiment. There are close relationships between this problem and transonic aerodynamics. The second problem is the study of nonlinear effects on gravitational waves in the Einstein field equations of general relativity, an issue of fundamental scientific significance. The third problem is the study of nonlocal, nonlinear equations for hyperbolic surface waves in elasticity and magnetohydrodynamics. Such waves arise, for example, in surface acoustic wave devices used in signal processing. The fourth problem is the study of the interaction of high frequency vorticity waves and mean flows in incompressible fluids. This research will describe the nonlinear development of vorticity instabilities, and is relevant to the closure problem for turbulent flows. The fifth problem is the propagation of fronts in a bistable oscillatory system of reaction-diffusion equation that provides a simplified model of the collective behavior of beta-cells in the pancreas, which produce insulin. The sixth problem is the study of surfactant deposition by a spreading liquid drop. This problem has industrial applications in the use of droplets for the deposition of surface films.
Waves, such as sound waves, elastic waves, and gravitational waves, are an important feature of many physical and biological systems. Large amplitude waves may behave nonlinearly, and this leads to new effects not seen in linear waves: for example, the generation of shock waves by an aircraft traveling at speeds close to or above the speed of sound. A detailed analysis of the equations that describe nonlinear waves is often very difficult. The aim of this research is an increased understanding of nonlinear waves in the context of a variety of applications that involve qualitatively interesting and poorly understood phenomena, and that are related to problems of current scientific and technological interest.

DMS-0135345
PI: Bruno L. Nachtergaele, UC Davis
Title: Research Focus Groups in Mathematics

Abstract

The UCDavis VIGRE project offers a range of activities designed to integrate the excitement of research mathematics into every facet of its undergraduate, graduate and postdoctoral programs. To accomplish this, the investigators and their colleagues are running up to four Research Focus Groups as a means to coordinate the activities in four research areas. Each Research Focus Group includes regular faculty members, postdoctoral fellows, graduate students, and undergraduates. The areas of research of the Groups vary from year to year, depending on the interests and expertise of the faculty and graduate students. Initial proposals include: 1) Geometry and Topology, including Computational Aspects; 2) Dynamics of Classical and Quantum Many-Body Systems; 3) Mathematical Biology, including Protein Structure and Function, Biofluid Mechanics, and Spatial Models in Ecology; 4) Discrete Mathematics, including Algebraic and Geometric Combinatorics and Stochastic Discrete Optimization. The Research Focus Group activities include a research seminar, a reading seminar, research projects, and outreach programs. An internship program has also been set up as part of the project.

Among the goals of the Research Focus Groups are to introduce undergraduates to mathematical research, to provide a mechanism for graduate students to interact closely with faculty and postdoctoral fellows early in their graduate careers, and to give postdoctoral fellows an opportunity to learn leadership skills necessary to a successful mathematical career. The Research Focus Groups offer participants direct involvement in active areas of research of great current interest. This allows beginning students to focus on a promising career path in mathematics early on, and creates an environment where they can be more effectively prepared for such careers. The VIGRE project and the Research
Focus Groups also serve as a focus of new initiatives in graduate recruitment. Feedback from Research Focus Groups and internship participants is used to revise the mathematics undergraduate and graduate programs at UC Davis.

ABSTRACT

FRG: Multi-Dimensional Problems for the Euler Equations
of Compressible Fluid Flow and Related Problems
in Hyperbolic Conservation Laws

Historically, fluid and solid mechanics study the motion of
incompressible and compressible materials, with or without internal
dissipation. For gases and solids with internal dissipation as a
secondary effect, the gross wave dynamics is governed by inviscid,
thermal diffusionless, dynamics. Within these categories, compressible
motion for solids corresponds to the study of elastic waves and their
propagation; compressible motion for fluids is usually associated with
inviscid gas dynamics. Furthermore both compressible solids and
fluids exhibit shock waves and hence we must search for discontinous
solutions to the underlying equations of motion.
Incompressible motion on the other hand concerns
itself with the motion of denser fluids where the idealization of
incompressibility is useful, e.g. water or oil, as well as the motion of
certain solids like rubber. While there are still many important
mathematical issues to be resolved for incompressible fluids, for example,
the well-posedness of the Navier-Stokes equations in three space
dimensions, the mathematical study of compressible
solids (as represented by the equations of nonlinear elastodynamics) and
fluids (as represented by the Euler equations of inviscid flows)
in two and three space dimensions is even less developed.
This provides the motivation to the proposers to collaborate in a
three year effort to advance the mathematical understanding of the
multi-dimensional equations of inviscid compressible fluid dynamics
and related problems in elastodynamics.
The core of our plan is to arrange a sustained interaction between and
around the members of the group, who will
(1) collaborate scientifically, focusing on the advancement of the
analysis of multi-dimensional compressible flows by developing new
theoretical techniques and by using and designing effective, robust and
reliable numerical methods;
(2) work together over the next several years to create the environment
and manpower necessary for the research on multi-dimensional compressible
Euler equations and related problems to flourish; and in the meantime,
(3) share the responsibility of training graduate students and
postdoctoral fellows.

The project is devoted to a mathematical study of the Euler equations
governing the motion of an inviscid compressible fluid and related
problems. Compressible fluids occur all around us in nature, e.g. gases
and plasmas, whose study is crucial to understanding aerodyanmics,
atmospheric sciences, thermodynamics, etc.
While the one-dimensional fluid flows are rather well understood, the
general theory for multi-dimensional flows is comparatively mathematically
underdeveloped. The proposers will collaborate in a three
year effort to advance the mathematical understanding of the
multi-dimensional equations of inviscid compressible fluid dynamics.
Success in this project will advance knowledge of this fundamental area of
mathematics and mechanics and will introduce a new generation of
researchers to the outstanding problems in the field.

Waves are a phenomenon of fundamental physical and technological importance. For example, the observation of waves from distant sources (whether gravitational waves generated by the collision of black holes or sound waves generated by a submarine) allows one to make useful conclusions about the nature of those sources, and the manipulation of waves has many technological applications (such as the use of ultrasonic surface acoustic waves in signal processing devices).





Nonlinear hyperbolic waves are an especially interesting and important class of waves. They include the shock waves generated by aircraft in transonic or supersonic flight, the seismic waves (and possible Tsunamis) generated by earthquakes, and the gravitational waves predicted by Einstein's general theory of relativity.





The dominant properties of hyperbolic waves are described mathematically by hyperbolic systems of partial differential equations. In a first approximation, hyperbolic waves are non-dispersive, meaning that the velocity at which they propagate, and their other properties, do not depend on the wavelength of the waves. This contrasts with dispersive waves, such as light waves in an optical fiber or Rossby waves in the atmosphere. The effects of nonlinearity on hyperbolic waves have a unique character; for example, the formation of singularities, such as shock waves, is a ubiquitous phenomenon, and one with significant physical implications.





The goal of the proposed research is to study nonlinear hyperbolic waves from a unified mathematical perspective in close connection with various physical applications. Specific topics include the reflection of shock waves and related questions in transonic fluid flows; the propagation of nonlinear hyperbolic surface waves, including Rayleigh (or surface acoustic) waves in elasticity; and the effect of nonlinearity on gravitational waves in general relativity.

The project uses the resources of the UC Davis Department of Mathematics to attract more US citizens to careers in the mathematical sciences, with special attention paid to women and underrepresented groups. The main tool for this is the Research Focus Group (RFG), a construct created in the prior VIGRE grant, which has been wildly successful. An RFG is a group of faculty, postdocs, graduate students, and undergraduate students, who collaborate in a year-long intense exploration of a particular research area. Through RFG activities, members are exposed to cutting-edge research, and they participate in all its aspects: exploration, discovery, leadership, exposition, and lecturing. In particular, graduate students develop strong communication skills, both written and verbal. Every year, four RFGs are selected and funded.

In addition to the RFGs are activities new to this project: A Mini-Grant Program through which students can apply not just for summer stipends but for conference travel funds, or even to organize workshops, to run outreach activities, or to host visiting mathematicians. Writing a mini-grant is not only a valuable challenge that forces the students to plan and organize a research project, but it also unleashes the creativity of the graduate students and postdocs.

Mathematics majors are recruited aggressively using several tools: The EXPLORE Math program is an outreach program for high school and undergraduate students that was created by VIGRE graduate students. The project supports local student organizations, Math Club and Math Cafe, to promote peer-based recruitment and retention of mathematics majors. Mathematics fairs for senior high school students are held in geographic proximity (which includes large communities of underrepresented minorities) to attract incoming freshmen to major in math and promote career paths that use mathematics. VIGRE-funded students and postdocs are required to participate in at least two of the many outreach activities.

The increased focus on particular research areas leads to increased research productivity among students and postdocs. It may also lead to new developments and results obtained by RFG participants (e.g., theses, publications, and lecture notes). Attending conferences increases the exposure of graduate students to new ideas and to the dissemination of research.

The impact on creating a larger and better group of U.S. citizens and permanent residents working in mathematics will be significant. The project directly affects the surrounding geographic area of Northern and Central California. In the recruitment plan special attention is given to attracting and retaining minorities currently underrepresented in mathematics but abundant in this geographic area.

The project addresses the mathematical modeling and analysis of nonlinear, nondispersive wave propagation in continuum mechanics. It focuses on waves modeled by nonlinear hyperbolic PDEs, and related equations, that propagate along boundaries or interfaces (such as discontinuities in vorticity, vortex sheets, material boundaries, and shock waves). These surface waves often display a complex nonlocal, nonlinear behavior which is not well-understood. The principal investigator will derive and study reduced asymptotic equations that describe these waves in a range of physical applications. A typical feature of the resulting nonlocal quasilinear equations is that they are Hamiltonian, and they may be expressed in both spectral and spatial forms, leading to connections with multilinear harmonic analysis. Fundamental questions concerning these equations include the life-span of smooth solutions, the formation and physical interpretation of singularities, and the global existence of weak solutions.

Surface waves are waves that propagate along a boundary or interface. Since they are guided along an interface, they decay more slowly than bulk waves, which explains why the surface seismic waves generated by an earthquake are the most destructive far from their source. Surface waves are widely used in technological applications, such as ultrasonic surface acoustic wave devices in cell phones or nanophotonic surface plasmon devices, because they are directly accessible to detection and manipulation. Small-amplitude waves are well-described by linear equations, but nonlinear effects become important at larger amplitudes and lead to qualitatively new phenomena such as the formation of singularities (for example, shock waves in a compressible fluid). Nonlinearity makes the mathematical analysis of these problems very challenging. An additional feature of surface waves is that the effects of nonlinearity may be nonlocal because what happens at one point on the surface can influence what happens elsewhere on the surface through the bulk medium. The principal investigator plans to study the fundamental qualitative properties of such nonlinear, nonlocal surface waves in the context of a wide variety of physical problems. The results will have potential applications in fluid dynamics, including transonic flow, elasticity, magnetohydrodynamics, geophysics, and condensed matter physics.

A combination of student understanding of the nature of science, an epistemology of science, the student's culture and general worldview has been shown to influence a student's success in science. The underlying scientific epistemological view (SEV) of the student may be a significant factor in student views about science as well as student's level of success in science. The proposed research will investigate the scientific epistemological views (SEVs) of STEM majors at HBCUs and other universities to determine to what extent student SEV is a factor in undergraduate students' experiences and subsequently retention in a STEM major to graduation.

The research will look for potential correlation between student's view of science as a desired major and student's SEV. In addition the study will investigate differences between student ethnic group classification and SEV in order to provide information about the diversity of epistemic thought in science classrooms.

The research methodology for this project consists of a quantitative and qualitative cohort study using a survey instrument referred to as the SEV-S to provide data regarding student's scientific epistemological views. A pilot study has been completed using this instrument and design methodology at Tennessee State University and statistically significant differences in epistemologies of science were found between STEM majors/non-STEM majors, male/female students, and students of different ethnicities.

This project addresses the mathematical modeling and analysis of nonlinear wave propagation in a variety of physical systems. It focuses on nondispersive waves, especially surface waves that propagate on interfaces such as discontinuities in vorticity, vortex sheets, material boundaries, water waves, and shock waves. Many of the wave motions considered in the proposal have constant, nonzero frequency in the linearized limit. These waves form a comparatively little studied class of nondispersive waves, and the proposed research aims to develop an understanding of their nonlinear dynamics, which is qualitatively different from that of dispersive waves or nondispersive hyperbolic waves. The proposed research will derive and study asymptotic descriptions of these waves and will also develop normal form transformations for quasi-linear wave equations. Hamiltonian dynamics provides unifying framework for most of the nonlinear wave motions to be studied in the proposed research. For small-amplitude waves, this description is more easily carried out in spectral form, which is particularly appropriate for the surface waves considered in the proposed research because of the spatial nonlocality of their interactions. The issue of understanding the relationship between the spectral and spatial descriptions of the resulting nonlinear dynamics is a fundamental one and one that is relevant to many other problems. A further topic of the proposed research is a study of the glancing Mach reflection of shock waves. Shock reflection is one of the most important multi-dimensional problems for hyperbolic conservation laws, leading to remarkably interesting and complex phenomena.These results should also shed light on related problems in transonic aerodynamics.

Surface waves are waves that propagate along a boundary or interface. The most familiar example consists of the water waves on the surface of a body of water, like an ocean. Another type of surface wave consists of the Rayleigh waves on a solid interface. These waves are generated by earthquakes, and they are also used in technological applications, such as ultrasonic surface acoustic wave devices in cell phones. A further example consists of the electromagnetic surface waves, or surface plasmons, on the interface between a metal and an insulator, which find applications in photonics. Small-amplitude waves are well-described by linear equations, but at larger amplitudes nonlinear effects become important. These effects lead to qualitatively new phenomena such as wave-breaking, the formation of shock waves or other singularities, and the generation of new waves by nonlinear wave-interactions. Nonlinearity, and the possibility of a free surface that moves with the wave, makes the mathematical analysis of these problems very challenging. An additional feature of surface waves is that the effects of nonlinearity may be nonlocal because what happens at one point on the surface can influence what happens elsewhere on the surface through the bulk medium. The principal investigator plans to study the fundamental qualitative properties of such surface waves in the context of a wide variety of physical problems. The results will have potential applications in fluid dynamics, transonic flow, elasticity, magnetohydrodynamics, geophysics, and condensed matter physics.

The addition of a hypocone, a fourth cusp, to the upper molar in mammals has been a common transition in dental evolution. Using suborder Microchiroptera (microbats) as a case study, the proposed research will examine the adaptive significance of the hypocone. This research will test whether the hypocone functions to increase tooth strength, by dissipating stress within the enamel to avoid tooth cracking. Finite Element Analysis, a computer simulation technique commonly used in engineering to assess how structures respond to physical loads, will be used to determine the strength attributes of teeth with and without hypocones. Ultimately, this work will examine how the hypocone?s role in tooth strength may have influenced the evolution of molar shape.

This project uses innovative computer modeling techniques to examine adaptation, a fundamental evolutionary concept. Facets of this work are applicable throughout the scientific community (from paleontologists studying the form and function of fossils to ecologists studying mammalian diversity patterns). Further, an understanding of adaptation is vital for anyone wanting to know how life has evolved. This work will encourage undergraduate participation in scientific research. All computer models will be submitted to the Morphobrowser and Digimorph online databases to allow access to both the scientific community and the public.

Surface waves are waves that propagate along a boundary or interface. They arise in many physical systems, such as water waves on the surface of the ocean and seismic surface waves generated by earthquakes. They also have a wide variety of technological applications, such as ultrasonic surface acoustic waves used in analog filters and surface plasmons used in optical and nano-optical devices. At lower intensities, effects of different waves add (waves superpose) and may be described by linear theories, but at higher intensities nonlinear effects become significant. Nonlinearity leads to qualitatively new phenomena, including wave interactions and the formation of singularities (such as shock waves). This project addresses the fundamental nonlinear dynamics of surface waves and their applications, especially nondispersive surface waves whose phase speed or frequency is independent of their wavelength. As a part of the project, graduate and undergraduate students will be trained through involvement in the research.

The project will carry out an asymptotic analysis of nonlinear surface waves in a variety of physical systems. Applications will be made to electromagnetic surface waves, or surface plasmons, waves on vorticity discontinuities and surface quasi-geostrophic fronts, plasma waves, and Rayleigh waves in elasticity. The resulting asymptotic equations are typically spatially nonlocal and quasilinear. These equations will be studied analytically and numerically, in particular with the aim of understanding the nonlinear focusing and possible formation and propagation of singularities in these surface waves. In addition, the project will study the diffraction and glancing Mach reflection of shock waves, together with related problems in transonic aerodynamics.

Fluids, such as air or water, support many different kinds of waves. For example, sound waves propagate through compressible fluids, water waves propagate on the surface of an ocean, and large-scale atmospheric waves affect the earth's weather and climate. At sufficiently low intensities, waves are modeled by linear systems of equations, which are relatively well-understood; but at higher intensities, the equations become nonlinear. Nonlinearity introduces fundamental mathematical difficulties and leads to new physical phenomena, such as the shock waves generated by supersonic aircraft, the solitons (particle-like waves) that propagate on shallow water, and nonlinear wave interactions that generate additional waves. The aim of this project is to study the dynamics of waves in fluids and uncover new nonlinear phenomena in the context of specific physical applications. This award will also provide support for the involvement of one graduate student in the project.

This project addresses the mathematical modeling and analysis of nonlinear wave propagation in a variety of fluid systems. It focuses on the asymptotic analysis of nonlinear, nondispersive and dispersive waves, including waves that propagate on boundaries, free surfaces, or interfaces. A unifying framework is provided by Hamiltonian dynamics, which is an essential structure for the waves considered in the proposed research. Physical applications include waves on vorticity discontinuities in incompressible fluids, temperature fronts in surface quasi-geostrophic flows, current-vortex sheets in plasmas, shock wave reflection, and the resonant interaction of sound and entropy waves. A number of these problems lead to common mathematical themes, such as degenerate dispersive shocks, normal form transformations for quasilinear wave equations, and the formation and propagation of singularities.

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.