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Integrability in Multidimensions and Boundary Value Problems

多维可积性和边值问题

基本信息

批准号：
EP/H04261X/1
负责人：
Athanassios Fokas
金额：
$ 59.28万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH04261X%2F1
关键词：
Integrability Multidimensions Boundary Value Problems

项目摘要

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.

存在某些独特的称为可积的非线性方程。这种特殊方程的数学分析的影响怎么估计都不为过。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，包括解的长期渐近性和孤子所起的中心作用。其次，很明显，由可积方程教授的一些课程即使在不可积的情况下也适用。事实上，在过去的十年里，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来自可积系统理论的结果，尽管是在不可积的环境下。可积方程理论中最重要的两个公开问题可能是(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组的推导和解。在过去的12年里，和平研究所在解决这两个问题方面取得了重大进展。也就是说，关于(A)他引入了分析二维边值问题的统一方法，关于(B)他在4+2中导出并求解了可积的非线性偏微分方程组。然而，许多基本问题仍然悬而未决。该建议旨在研究几个这样的问题，其中最重要的是：(A)将求解边值问题的方法从二维推广到三维；(B)将新的可积偏微分方程组从4+2减少到3+1。此外，还将研究著名的Sine Gordon方程的Camassa-Holm模拟，椭圆型Ernst方程的几个边值问题，以及具有时间周期边界条件的半直线上的KdV方程。