Integrable Nonlinear Evolution PDEs on the Interval

在Deconinck教授到剑桥进行为期三个月的研究访问期间，我们将研究在有限区间上著名的非线性Schr\ odinger方程（NLS）的分析中出现的两个问题：(a)在边界值在空间上是周期性的特殊情况下，求解有限区间上初边值问题的一般理论的约简。(b)给定Dirichlet边界条件在时间上是周期的情况下解的渐近性质。关于第一个问题，我们注意到相关的优美的代数-几何理论用函数来描述特定的有限间隙情况。我们的目标是用矩阵黎曼-希尔伯特问题来描述一般情况，并作为特殊情况重新推导有限间隙表示的结果。关于第二个问题，我们注意到，对于NLS的线性化版本，最近已经证明解的渐近行为取决于边界数据的时间周期$T$与$L^2/ \pi$的可通约性，其中$L$是有限区间的长度。我们的目标是获得NLS的类似结果。

During the three-month research visit of Professor Deconinck to Cambridge, we will investigate two problems which arise in the analysis of the celebrated nonlinear Schr\ odinger equation (NLS) formulated on a finite interval: (a) The reduction of the general theory for solving intial-boundary value problems on the finite interval in the particular case that the boundary values are periodic in space. (b) The asymptotic behaviour of the solution in the case that the given Dirichlet boundary conditions are periodic in time. Regarding the first problem, we note that the related beautiful algebraic-geometric theory characterises the particular finite-gap case in terms of theta functions. Our goal is to characterise the general case in terms of a matrix Riemann-Hilbert problem and to rederive the results of the finite-gap representation as a particular case. Regarding the second problem, we note that for the linearized version of the NLS it has been shown recently that the asymptotic behaviour of the solution depends on the commensurability of the time period $T$ of the boundary data with $L^2/ \pi$, where $L$ is the length of the finite interval. Our goal is to obtain an analogous result for the NLS.