The Semiclassical Limit of the Focusing Nonlinear Schroedinger Equation

NSF Award Abstract - DMS-0103909 Mathematical Sciences：The Semiclassical Limit of the Focusing Nonlinear Schroedinger Equation摘要0103909米勒本项目研究聚焦非线性薛定谔方程在奇异半经典极限下的解的行为，特别关注与给定初始数据相关联的解。 具体的研究目标包括：（i）推广矩阵Riemann-Hilbert问题的最速下降过程，以恢复一般实解析和解析初始数据的初值问题的渐近性，（ii）计算非自伴Zakharov-Shabat算子的严格谱渐近性，并考虑逆散射问题中的误差估计，（iii）研究上半平面中的最小加权绿色容量集，并将它们与半经典渐近性牢固地联系起来，以及（iv）确定渐近性对数据中奇异点的存在的敏感性以及对结构扰动的鲁棒性。 分析将采用数值方法，仔细渐近谱分析的一个家庭的非自伴微分算子，和潜在的理论方面的功能和复杂的analysis.The聚焦非线性薛定谔方程是一个普遍存在的模型方程的传播波的许多不同种类（水波，光波等）。在同时存在的非线性效应，可以“自我放大”的波和“色散”，可以把波分开。 特别是，它是一个测试和接受的模型传输光波脉冲沿着某些类型的玻璃光纤。 这个项目将产生新的理解，这个模型方程的色散项的系数在方程中的情况下，是相对较小的，或者，非线性过程占主导地位的广泛的扰动的演变短的时间。 目前安装在许多现代电信系统中的“色散位移”光纤提供了一种环境，其中色散和非线性的影响正好以这种偏斜的比例存在。 该项目的结果将可能影响下一代高速光通信系统的分析和设计。

NSF Award Abstract - DMS-0103909Mathematical Sciences: The Semiclassical Limit of the Focusing Nonlinear Schroedinger EquationAbstract0103909 MillerThis project addresses the behavior of solutions of the focusing nonlinear Schroedinger equation in the singular semiclassical limit, with particular attention paid to solutions that are tied to definite given initial data. Specific research goals include (i) generalizing a steepest-descents procedure for matrix Riemann-Hilbert problems to recover asymptotics of the initial-value problem for general real-analytic and oscillatory-analytic initial data, (ii) computing rigorous spectral asymptotics for the nonselfadjoint Zakharov-Shabat operator and taking estimates of the error into account in the inverse-scattering problem, (iii) studying sets of minimal weighted Green's capacity in the upper half-plane and relating them solidly to semiclassical asymptotics, and (iv) determining the sensitivity of the asymptotics to the presence of singularities in the data and also robustness to structural perturbations. The analysis will employ numerical methods, careful asymptotic spectral analysis of a family of nonselfadjoint differential operators, and potential-theoretic aspects of functional and complex analysis.The focusing nonlinear Schroedinger equation is a ubiquitous model equation for the propagation of waves of many different kinds (water waves, light waves, etc.) in the simultaneous presence of nonlinear effects that can "self-amplify" the waves and "dispersion" which can pull the waves apart. In particular, it is a tested and accepted model for the transmission of lightwave pulses along certain types of glass optical fibers. This project will produce new understanding of this model equation relevant to situations where the coefficient of the dispersive term in the equation is relatively small, or alternatively, nonlinear processes dominate the evolution of broad disturbances for short times. "Dispersion-shifted" optical fibers currently being installed in many modern telecommunication systems provide an environment where the effects of dispersion and nonlinearity are present in precisely such a skewed proportion. The results of this project will be likely to influence the analysis and design of the next generation of high-speed optical telecommunications systems.