Analytical Study of Phase-Sensitive Solitary Wave Behavior

[804814 .米勒]本研究涉及某些不可积非线性薛定谔型偏微分方程的孤立波（脉冲）解的行为，这些方程近年来引起了物理学家和工程师的注意，特别是在非线性光学应用中。研究将主要集中在二次非线性材料的参数波导中脉冲传播模型的系统。这些方程有一个共同的特性，即它们的解表现出高度相敏的行为（例如在碰撞结果中）。研究的一部分将使用变分技术和线性化算子的详细特征值分析来研究某些系统的行波的存在性和稳定性。研究的第二部分将涉及发展常微分方程作为相敏脉冲行为的模型，包括两个二次脉冲之间的碰撞。要研究的方程出现在光纤中；他们的解决方案模拟光脉冲在光纤中传播。光学科学家和工程师对它们很感兴趣，因为它们模拟的光脉冲非常窄，产生光脉冲所需的能量也非常低；希望是许多这样的脉冲可以紧密地聚集在一起，以提高通过光纤传输数据的速度。这项研究将通过确定预测这些脉冲稳定性的重要因素（当脉冲受到噪声和与光纤和其他脉冲的相互作用的影响时，它在多大程度上保留了传输信息所需的脉冲形状？）和它们在碰撞中的行为（当两个脉冲迎面相遇时）来解决这些脉冲的基本特性。

9804814MillerThis study concerns the behavior of solitary-wave (pulse) solutions to certain nonintegrable Nonlinear Schrodinger-type partial differential equations that have drawn attention from physicists and engineers in recent years, particularly in nonlinear optics applications. The study will focus mainly on systems that model pulse propagation in parametric waveguides made of materials with quadratic nonlinearity. These equations share the property that their solutions show highly phase-sensitive behavior (for example in collision outcomes). One part of the study will investigate the existence and stability of travelling waves for certain systems, using variational techniques and detailed eigenvalue analysis of linearized operators. A second part of the study will involve the development of ordinary differential equations as models for phase-sensitive pulse behavior, including collisions between two quadratic pulses.The equations to be studied arise in fiber optics; their solutions model light pulses travelling down a fiber. They are of interest to optical scientists and engineers because they model light pulses that are unusually narrow and require unusually low energies to produce; the hope is that many such pulses could be packed closely together to increase the speed of data transmission through optical fibers. This study will address fundamental properties of these pulses by identifying factors important for predicting their stability (when a pulse is affected by noise and interactions with the fiber and other pulses, to what extent does it retain the pulse shape necessary to transmit information?) and their behavior in collisions (when two pulses meet each other head-on).