RUI: Mathematical Analysis of Several Models in Nonlinear Optics

非线性光学的最新进展包括超短光脉冲的产生和应用，其持续时间通常为飞秒（一千万亿分之一秒）的数量级。这些脉冲的持续时间接近基本原子和分子过程的时间尺度。它们传输能量的速度如此之快，以至于可以在不损坏生物结构的情况下探测生物结构，并以最小的热效应在微米尺度上对材料进行修改。例如，飞尺度激光器用于显微外科手术或材料加工。对于超短脉冲，用于较长光脉冲的传统数学模型非线性Schrödinger （NLS）和耦合非线性Schrödinger （CNLS）方程失去了预测能力。本研究项目涉及两个新颖的数学模型：复短脉冲方程和耦合复短脉冲方程。本课题致力于超短光脉冲描述方程的解的分析和计算。所研究的模型是可积的；该项目涉及CSP方程，CCSP方程及其可积半离散类似物的研究，其中空间变量在晶格中取值，而结果方程的可积性保持不变。具体而言，研究者计划：(1)构造半离散CSP方程，并推导出包括亮孤子、暗孤子和类流氓波解在内的精确解；(2)构造CCSP方程的亮-亮、暗-暗、亮-暗（混合）孤子解以及异常波解；(3)利用半离散CSP方程作为一种新的自适应运动网格数值方法来模拟连续CSP模型的解。作为全国第二大西班牙裔服务机构，研究者努力激励和鼓励本科生和研究生从事应用数学的前沿研究。

The most recent advances in nonlinear optics include the generation and applications of ultrashort optical pulses, whose time duration is typically of the order of femtoseconds (one quadrillionth of a second). Duration of these pulses is approaching the timescales of fundamental atomic and molecular processes. They deliver energy so quickly that it allows them to probe living structures without damage and to make material modifications on the micron scale with minimal heat effects. Femtoscale lasers are used, for example, in microsurgery or materials processing. For ultrashort pulses, the traditional mathematical models used for longer optical pulses, the Nonlinear Schrödinger (NLS) and Coupled Nonlinear Schrödinger (CNLS) equations, lose their predictive capabilities. This research project is concerned with two novel mathematical models: the Complex Short Pulse (CSP) equation and the Coupled Complex Short Pulse (CCSP) equation. The project is devoted to analysis and computation of solutions of these equations for the description of ultrashort optical pulses.The models to be investigated are integrable; the project concerns the investigation of the CSP equation, the CCSP equation and their integrable semi-discrete analogues, where the spatial variable takes values in a lattice, while the integrability of the resulting equations is preserved. In particular, the investigator plans to: (1) construct the semi-discrete CSP equation and derive exact solutions including bright and dark solitons and rogue-wave-like solutions; (2) construct bright-bright, dark-dark and bright-dark (mixed) soliton solutions, as well as rogue wave solutions, of the CCSP equation; (3) utilize the semi-discrete CSP equation as a novel self-adaptive moving mesh numerical method for simulating solutions of the continuous CSP model. Being at the second largest Hispanic-serving institution in the country, the investigator strives to motivate and encourage undergraduate and graduate students to engage in cutting-edge research in applied mathematics.