Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - BOPNIE (Boundary value problems for nonlinear integrable equations)

Teaser

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...

Summary

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.

The goal of the present project is to solve several open problems related to boundary value problems for integrable PDEs. For example, we aim to answer some long-standing open questions related to the so-called wavemaker problem, which consists of computing the wave-shapes created by a paddle wavemaker at the end of a long wave tank. An understanding of this problem has implications for our understanding of the propagation and modeling of water waves. Of particular interest is the computation of asymptotic properties such as the long-time behavior. Another objective is to analyze the stationary axisymmetric Einstein equations with boundary conditions corresponding to a disk rotating around a central black hole. The computation of solutions of this type will help understand the curvature of spacetime around such objects and thus enhance our understanding of certain astronomical events. We also aim to solve a boundary value problem for the Einstein equations describing two colliding gravitational waves plane waves.

Work performed

One of the results achieved so far in this project is the effective solution of the boundary value problem for the Einstein equations corresponding to two colliding gravitational plane waves. This problem can be described mathematically by a Goursat problem for the so-called hyperbolic Ernst equation in a triangular domain. In a recent work, we use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann-Hilbert problem.

Another equation which we have studied extensively is the so-called sine-Gordon equation. This equation has numerous applications: It is the Gauss-Codazzi equation for surfaces of constant negative curvature embedded in three-dimensional space, it is the continuous limit of the Frenkel-Kontorova model in condensed matter physics, it models the magnetic flux propagation in Josephson junctions, and it can be used to describe several phenomena in nonlinear optics such as self-induced transparency. In this project, we have completed a large part of a program for the sine-Gordon equation envisioned in the 1990s by Cheng, Venakides & Zhou, which sought for the calculation of the solution and its winding number in the presence of both radiation and solitons.
We derive formulas which describe what the solution looks like for large times and investigate the effect of a boundary.

In another recent work, large time asymptotic formulas for another equation which models the propagation of certain types of plasma waves - the derivative nonlinear Schrödinger equation - are presented. We have also studied the so-called defocusing nonlinear Schrödinger equation. We have positively answered a conjecture claiming the absence of solitons for the defocusing NLS equation on the half-line. Furthermore, by performing a spectral analysis of the underlying differential equation, we have analyzed time-periodic boundary conditions. Using related techniques, we have derived an exact solution of the stationary axisymmetric Einstein equations corresponding to a certain type of rotating disk.

Final results

By developing new techniques for the analysis of boundary value problems for integrable equations, we have been able to solve and analyze several boundary value problems which were heretofore not solvable. To name a few examples, important progress has been made on problems with time-periodic boundary conditions, for the Einstein equations, and for integrable equations with higher order Lax pairs. We expect the progress to continue and that we will find new important results, for example, for the Einstein equations corresponding to stationary axisymmetric spacetimes, for integrable equations in higher dimensions, and for spatially periodic solutions.