This project will investigate the long-term behavior of solutions to dispersive Hamiltonian partial differential equations, such as the semilinear wave, Klein-Gordon, and nonlinear Schroedinger equations. These equations can be either defocusing or focusing, which distinguishes whether the nonlinearity is attractive or repulsive. In the latter case, one typically encounters various regimes depending on the power of the nonlinearity, which allows for rich dynamics ranging from long-term existence and dispersion, to finite-time blowup. Recently, in joint work with Kenji Nakanishi from Kyoto University, Japan, the principal investigator has given a complete characterization of all possible dynamics at energies close to the ground state energy for a large class of these focusing dispersive wave equations. This classification is achieved by a combination of dynamical systems methods (hyperbolic dynamics, invariant manifolds), with partial differential equations arguments such as concentration compactness and the Kenig-Merle theory. Several important open problems remain, among which is to obtain this type of classification for the energy critical nonlinear wave equation. This is particularly relevant in view of the related but complementary research by Duyckaerts-Kenig-Merle on focusing equations. A long-term goal is to establish the soliton-resolution conjecture. This conjecture can be viewed as the nonlinear analogue of the celebrated asymptotic completeness property of the linear Schroedinger evolution.
Nonlinear wave equations play a central role in science. Maxwell's equations of electrodynamics are of this type, and they are arguably the most influential partial differential equations of modern science -- the existence of radio waves, and general electromagnetic radiation such as light and X-rays was predicted in the 1870s by Maxwell based on these equations alone and confirmed by experiment later that century. Needless to say, it is unthinkable to remove radio transmission, X-rays, lasers, microwaves, and many other electromagnetic radiation fields from our daily lives. In addition, Maxwell's wave equations have had theoretical impact far beyond anything of which nineteenth-century physicists and mathematicians could have conceived. Indeed, they make the constancy of the speed of light most natural, and the symmetries of Maxwell's system lead directly to Einstein's theory of special relativity. Unifying the latter with gravity then led to the general theory of relativity. In addition, quantum theory has provided many more examples of wave equations, in many cases nonlinear ones. For example, special solutions that go by the name of "dispersion managed solitons," and that solve a certain class of nonlinear Schroedinger equations arising in nonlinear optics, are indispensable today for the transmission of the world's internet traffic through carefully designed glass fiber cables. The introduction of these special cables, which consist of alternating stretches of different materials, drastically reduced transmission errors and cost, and allowed for a huge increase in the data volume being transmitted. This project focuses on the further development and study of nonlinear wave equations of the type that arise in many areas of physics and engineering. Time and time again, mathematicians have laid the foundations through pure research without which the engineering applications that profoundly affect our daily lives could not have been accomplished.
