Integrability in Multidimensions and Boundary Value Problems

存在某些特殊的非线性方程，称为可积方程。这种特殊方程的数学分析的影响不能被高估。例如，首先，对于可积方程，我们已经了解了解的行为的详细方面，其中包括解的长时间渐近性和孤子所扮演的核心角色。第二，很明显，可积方程的一些经验甚至在不可积的情况下也适用。事实上，在过去的十年中，许多关于适当的Sobolev空间中的偏微分方程适定性的研究都起源于可积系统理论的结果，尽管是在不可积的环境中。也许在可积方程理论中两个最重要的公开问题是（a）与初值问题相对的初边值问题的解和（B）3+1维可积非线性偏微分方程的推导和求解。在过去的12年里，PI在解决这两个问题方面取得了重大进展。也就是说，关于（a），他介绍了一个统一的方法，用于分析边界值问题的两个层面和关于（B），他推导并解决了可积非线性偏微分方程在4+2。然而，许多基本问题仍然悬而未决。该建议的目的是调查几个这样的问题，其中最重要的是：（a）从二维到三维的边值问题的解决方法的扩展和（B）减少新的可积偏微分方程从4+2到3+1。此外，著名的正弦戈登方程的Camassa-Holm模拟，几个边值问题的椭圆形版本的Ernst方程，和KdV方程的半线与时间周期边界条件也将被调查。

There exist certain distinctive nonlinear equations called integrable. The impact of the mathematical analysis of such special equations cannot be overestimated. For example, firstly, for integrable equations we have learned detailed aspects of solution behaviour, which includes the long-time asymptotics of solutions and the central role played by solitons. Secondly, it has become apparent that some of the lessons taught by integrable equations have applicability even in non-integrable situations. Indeed, many investigations in the last decade regarding well-posedness of PDEs in appropriate Sobolev spaces have their genesis in results coming from the theory of integrable systems, albeit in non-integrable settings. Perhaps the two most important open problems in the theory of integrable equations have been (a) the solution of initial-boundary as opposed to initial value problems and (b) the derivation and solution of integrable nonlinear PDEs in 3+1 dimensions. In the last 12 years, the PI has made significant progress towards the solution of both of these problems. Namely, regarding (a) he has introduced a unified approach for analysing boundary value problems in two dimensions and regarding (b) he has derived and solved integrable nonlinear PDEs in 4+2. However, many fundamental problems remain open. The proposal aims to investigate several such problems among which the most significant are: (a) the extension of the method for solving boundary value problems from two to three dimensions and (b) the reduction of the new integrable PDEs from 4+2 to 3+1. In addition, the Camassa-Holm analogue of the celebrated sine Gordon equation, several boundary value problems of the elliptic version of the Ernst equation, and the KdV equation on the half-line with time periodic boundary conditions will also be investigated.