Schr̈odinger Evolution Equations

From the outset, mathematicians and physicists have been concerned with the ways in which solutions to this equation can be associated to classical particle motion, and the ways in which they cannot: the classical dynamical behavior of particles is intermixed with dispersive spreadingand interference phenomena in quantum theory. More recently,nonlinearSchr̈odinger equations have come to play an essential role in the study of many physical problems. Nearly monochromatic waves with slowly varying amplitude occur frequently in science and technology. Second order expansions of physical models of wave phenomena around such waves lead naturally to the cubic nonlinear Schr ödinger equation. Thus, the nonlinear Schr ödinger equation is a canonical wave model since it emerges ubiquitously in the study of waves. Nonlinear Schr ödinger (NLS) equations appear in such diverse fields as nonlinear optics, superconductivity, oceanography, and quantum field theory. The main themes of research discussed at this workshop concern the Cauchy or initial value problem for Schr̈ odinger equations. Theoretical and applied aspects of nonlinear Schr ödinger equations are nicely surveyed in the textbooks [4], [17]. Nonlinear Schr̈ odinger evolutions involve a dynamical balance between linear dispersive spreading of the wave andnonlinear self-interactionof the wave. Generalizations of the physically relevant equations with nonlinear and dispersive parameters have been introduced to probe the interplay between these effects. For example, the semilinear initial value problem { i∂tu + ∆u = ±|u|p−1u u(0, x) = u0(x), x ∈ R, (1)