Periodic Reporting for period 4 - BOPNIE (Boundary value problems for nonlinear integrable equations)
Periodo di rendicontazione: 2020-11-01 al 2022-02-28
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.
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.
One of the results achieved 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 boundary value problem for the so-called hyperbolic Ernst equation in a triangular domain. By using the integrable structure of the Ernst equation, we were able to present the solution of this problem. Another equation which we have studied extensively is the sine-Gordon equation. This equation has numerous applications: It describes surfaces of constant negative curvature embedded in three-dimensional space, it models the magnetic flux propagation in Josephson junctions, and it can be used to describe phenomena in nonlinear optics. In this project, we have completed a large part of a program for the sine-Gordon equation envisioned by other researchers in the 1990s, which sought for the calculation of the solution and its winding number in the presence of both radiation and solitons. We have derived formulas which describe the solution for large times and we have investigated the effect of a boundary.
Large time asymptotic formulas for an equation which models the propagation of certain types of plasma waves - the derivative nonlinear Schrödinger equation - have also been derived. Furthermore, we have studied the nonlinear Schrödinger equation extensively – this is a famous nonlinear partial differential equation with multiple applications. For example, we have positively answered a conjecture claiming the absence of solitons for the defocusing nonlinear Schrödinger equation on the half-line, and we have constructed solutions and analyzed the long-time behavior of solutions of the focusing nonlinear Schrödinger equation. Methods for solving problems with time-periodic boundary conditions for this and other integrable equations have been developed. Another main result of the project is the introduction of a new approach for the study of space-periodic solutions of the nonlinear Schrödinger and other integrable equations. Tools have also been developed for the analysis of integrable equations with higher order Lax pairs. This has, among other things, led to the resolution of open questions related to the propagation of water waves.
The obtained results have been disseminated through international conferences, workshops, and seminars.
Large time asymptotic formulas for an equation which models the propagation of certain types of plasma waves - the derivative nonlinear Schrödinger equation - have also been derived. Furthermore, we have studied the nonlinear Schrödinger equation extensively – this is a famous nonlinear partial differential equation with multiple applications. For example, we have positively answered a conjecture claiming the absence of solitons for the defocusing nonlinear Schrödinger equation on the half-line, and we have constructed solutions and analyzed the long-time behavior of solutions of the focusing nonlinear Schrödinger equation. Methods for solving problems with time-periodic boundary conditions for this and other integrable equations have been developed. Another main result of the project is the introduction of a new approach for the study of space-periodic solutions of the nonlinear Schrödinger and other integrable equations. Tools have also been developed for the analysis of integrable equations with higher order Lax pairs. This has, among other things, led to the resolution of open questions related to the propagation of water waves.
The obtained results have been disseminated through international conferences, workshops, and seminars.
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 beyond the state of the art has been made on problems with time-periodic boundary conditions, problems with space-periodic boundary conditions, and problems for integrable equations with higher order Lax pairs.