Several of the most important partial differential equations (PDEs) in mathematics and physics are integrable. For example, integrable equations arise in the study of water waves, optical fibers, rotating galactic disks and stars, dynamical systems, gravitational waves, knot theory, plasma waves, and statistical mechanics. The purpose of this project is to develop new methods for solving boundary value problems for nonlinear integrable PDEs.
It was discovered in the 1960s that integrable equations can be analyzed via the so-called Inverse Scattering Transform. Given some initial data at time t = 0, the Inverse Scattering Transform provides a way to construct the solution at all later times t > 0, that is, to solve the initial-value problem for the equation. The introduction of this method was one of the most important developments in the theory of nonlinear PDEs in the 20th century. However, in many (perhaps most) laboratory and field situations, the solution is generated by what corresponds to the imposition of boundary conditions rather than initial conditions. Thus, for many years, one of the most outstanding open problem in the analysis of these equations was the solution of boundary value problems, or initial-boundary value problems, instead of pure initial-value problems. Although progress was minimal for a long time, several breakthroughs have occurred in recent years. This has opened up multiple avenues for groundbreaking research. It appears that a plethora of physically and mathematically important problems can now be solved for the first time.
In the present project, we have solved several open problems related to boundary value problems for integrable PDEs. For example, some long-standing open questions related to the propagation of water waves have been answered, and a new method for the analysis of integrable PDEs with periodic boundary conditions has also been developed. Moreover, new methods for the evaluation of asymptotic properties of solutions have been introduced and applied to a wide range of problems, including the propagation of fiber optic waves. As another example, a method to describe the collision of two plane gravitational waves in Einstein's theory of relativity has been put forward. New asymptotic results for several integrable models in statistical physics have also been derived.
