Integrable Nonlinear Evolution PDEs on the Interval

在Deconinck教授对剑桥为期三个月的研究访问期间，我们将研究在分析有限区间上著名的非线性薛定谔方程(NLS)时出现的两个问题：(A)在空间中边值是周期的特殊情况下，有限区间上求解初边值问题的一般理论的简化。(B)在给定的Dirichlet边界条件是时间周期的情况下，解的渐近行为。对于第一个问题，我们注意到相关的优美的代数几何理论用theta函数来刻画特殊的有限间隙情形。我们的目标是用矩阵Riemann-Hilbert问题来刻画一般情况，并将有限间隙表示的结果作为特殊情况重新推导。对于第二个问题，我们注意到，对于NLS的线性化版本，最近已经证明了解的渐近行为取决于边界数据的时间周期$T$与$L^2/\pi$的可公度性，其中$L$是有限区间的长度。我们的目标是为NLS获得类似的结果。

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.