Abstract of Proposed Research Wilhelm Schlag
This project is explore the long time behavior, or prove that singularities will form in finite time, of solutions of equations that arise in mathematical physics. The equations under consideration typically admit nonlinear bound states (solitons or instantons) and much research has recently been devoted to the perturbative analysis of such solutions. It is well known that solutions of nonlinear Schroedinger and wave equations of the focusing type may blow up in finite time (if the energy of the data is negative, for example). It turns out, however, that global solutions exist if the data belong to a submanifold of finite co-dimension (a "center-stable" manifold in the language of dynamical systems). We shall investigate whether there is a manifold that divides a region of blow-up from one of scattering. Our goal is to obtain a deeper understanding of blow-up phenomena. Recently, progress was made for the critical wave-map equation into the two-dimensional sphere with regard to blow-up. It can be shown in a very precise and quantitative way that blow-up for this equation occurs through the bubbling off of energy via a non-constant harmonic map. Moreover, it turns out that the blow-up rate can be prescribed a priori. Similar phenomena occur for the semi-linear energy critical focusing equation in three plus one dimensions. Currently we do not understand which classes of equations admit this kind of phenomenon.
Much of the success of science and engineering lies with its effective use of mathematical tools, both in terms of modeling and for computational simulation. The nonlinear Schroedinger equation arises in various applications in optics where a bound state (soliton) for represents a particle, or beam, that travels for a long time without disintegrating. An important issue is to understand the stability or instability of these solitons. That is, whether they persist under small perturbations or not? The theoretical understanding of these issues is very difficult, and is requiring new insights into mathematical problems. This project will investigate these problems and develop methods that may be used by practicing scientists and engineers.
The methods of mathematics pervade all of the sciences and engineering. Since Isaac Newton's times we have expressed how natural processes evolve in time through differential equations. While classical mechanics which deals with the movement and intercation of massive bodies, such as planets around the sun, is cast in terms of ordinary differential equations, the evolution of fluids for example is described by a very difficult system of partial differential equations. This latter type of differential equation expresses the evolution of a continuum of points, i.e., a collection of infinitely many points, in such a way that the laws which determine the evolution depend on how the points are distributed in space. This latter distribution changes, over time, so that a complicated interaction is established that is very hard to understand - which means it is also very hard to predict. An excellent example of this situation is furnished by the weather. All physical theories of the past 150 years such as Maxwell's electromagnetic equations, the theory of gravity (dealing with very large scales), and quantum mechanics (dealing with very small scales) are all formulated in terms of such partial differential equations. Many processes in nature evolve through waves, such as sound and light or any kind of electromagnetic radio wave. The PIs research focuses on the exact understanding of solutions to nonlinear wave equations. This type of equation is of basic importance to the understanding of transmission of signals, such as through cables. Somewhat amazingly, it is also essential to the description of the evolution of elementary particles - the basic building blocks of nature. Given an initial state of a system, one of the most basic as well as most difficult, questions is to descibe the state of the system after a comparatively long time. The predicition of the weather being a typical example. The PI has developed methods by which certain general statements can be made about the long-term behavior of waves which satisfy a deceptively simple type of wave equation. The difficulty lies with the property of waves to exhibit self-interactions, which may lead to the complete break-down of the wave. This can occur in nature, often with detrimental results such as the melt-down of optical materials through which laser light is being transmitted. In recent years, important advances on the rigorous understanding of such phenomena were made. These were facilitated by applying tools and ideas from diverse areas of mathematics such as harmonic analysis, dynamics sytems, and partial differential equations. While we have had some measure of success, much remains to be done. The impact of mathematics on society is large and profound, even though it may not be that visible on first sight. Today aerospace industry relies on the modeling of complicated flows around airplane fuselages on the computer, rather than in the wind tunnel. That this can be done at all is due to mathematicians who studied nonlinear wave equations rigorously and worked out how to compute solutions approximately on a computer. This can be very difficult to implement, and requires much skill and time. The field of numerical analysis is entirely devoted to this task. Too few young people have the desire or stamina to persevere in the study of mathematics or the sciences in general. In comparison to other developed as well as developing nations, the US exhibits a very small percentage of students eager to enter science and engineering. The training of the few who do is therefore of utmost importance. The PI is actively involved in the recruitment and training of young talent, and the fostering of interest and excitement amongst the youth for the rigorous arts that science is.
