Stability of Solitary Waves in Dynamical Systems

非线性波动方程为光学、流体动力学和各种其他物理系统中的许多现象提供了数学描述。这项研究的目的是对特殊和非常重要的孤立波解进行数学研究，这些孤立波解表示通过介质的单个波或单个波列。这种数学性质的解对于预测行为和为广泛的物理无关现象设计工程设备是很有用的，例如水波动力学，特别是巨大的海浪(“无赖”波)、材料的磁化、光在光纤和其他光学介质中的传播，或者一些量子力学动力学。即使不可能完全确定这些方程的解，通过考虑计算或近似解，通常也能令人满意地理解相应物理系统的演化。这是可行的，只要当不可避免的计算误差或参数中的不确定性引入时，解的行为不会发生质的变化。在数学上，这种对微小扰动的不敏感被称为“稳定性”。这个研究项目的主要目的是开发新的数学工具来研究非线性色散波及其孤立波解的稳定性。特别是，首席调查员(PI)的目标是获得关于长时间(渐近)行为的准确、定量的信息。研究生将通过参与这个项目得到培训和指导。PI将考虑物理应用中出现的各种模型的孤子稳定性问题，如行波、驻波、行波扭结。特别值得关注的是狄拉克方程、奥斯特罗夫斯基方程、短脉冲方程以及各种水波方程的接近孤子行为。此外，PI还将解决空间离散色散系统理论中的几个突出问题，即离散的非线性薛定谔方程和具有赫兹相互作用的颗粒链模型。这些都是模式，预计会出现新的范式。更准确地说，这些模型给他们的研究带来了障碍，这些障碍要么没有出现在相应的“标准”连续限制中，要么解决方案的行为方式与相应的连续类似物有很大不同。因此，需要开发新的数学方法来解决与研究这些离散模型的演化相关的挑战。

Nonlinear wave equations give a mathematical description for many phenomena in optics, fluid dynamics, and a variety of other physical systems. This research is aimed at the mathematical study of special and very important "solitary-wave" solutions that represent a single wave or a train of single waves traveling through the medium. Solutions of this mathematical nature are instrumental for predicting behavior and designing engineering devices for the wide range of physically unrelated phenomena, such as water wave dynamics, in particular, gigantic ocean waves ("rogue" waves), magnetization of materials, propagation of light in optical fibers and other optical media, or some quantum mechanical dynamics. Even when it is not possible to completely determine the solutions to such equations, the evolution of the corresponding physical system can often be understood satisfactorily by considering a computational or approximate solution. This is feasible, provided the behavior of solutions does not change qualitatively when unavoidable computational errors or uncertainty in the parameters are introduced. In mathematics this insensitivity to small perturbations is termed "stability." The principal goal of this research project is developing new mathematical tools for study nonlinear dispersive waves and stability of their solitary-wave solutions. In particular, the Principal Investigator (PI) aims at obtaining precise, quantitative information about the long time (asymptotic) behavior. Graduate students will be trained and mentored through their participation in this project.The PI will consider the questions of stability of solitons, such as traveling waves, standing waves, traveling kinks, for various models arising in physical applications. Of particular concern will be the close-to-soliton behavior of the Dirac equations, the Ostrovsky and the short pulse equations, as well as various water wave equations. In addition, the PI will address several outstanding problems in the theory of spatially discrete dispersive systems, of the type of the discrete nonlinear Schroedinger equation and the granular chain model with Hertzian interactions. These are the models, where new paradigms are expected to emerge. More precisely, the models present obstacles to their investigation that are either not present in the corresponding "standard" continuous limits or else, the solutions behave in a substantially different ways than the respective continuous analogues. Thus, new mathematical methods need to be developed to address the challenges associated with the study of evolution of these discrete models.