Stability in nonlinear wave equations

本项目研究非线性波动方程解的长时间行为。它的重点是寻找最小假设下的孤波解Schrödinger， Hartree，和Klein-Gordon方程是渐近稳定的。此外，工作将研究解分解成孤立波和散射波（渐近完备性），以及这些波之间的能量传递。该研究不仅将对形状随时间变化的孤立波的线性化所产生的半群算子进行新的色散估计，而且还将发展数学技术，将这些估计转化为完整非线性方程的长时间动力学结果。波动方程在量子力学、统计物理、光学和声学中无处不在，它们为流体动力学、力学和生物学中的某些现象建模。非线性波动方程支持孤立波，孤立波是模拟玻色-爱因斯坦凝聚、通过光纤传播信息和海啸形成等多种现象的基石。这些物理系统的详细动力学仍未完全了解。该项目旨在为这些现象建立一个数学理论，更广泛地说，为孤立波的长时间传播和相互作用建立一个数学理论。它将侧重于统计物理和非线性光学的特定模型，但该理论预计将具有足够的鲁棒性，可以应用于其他科学和工程领域中由模型产生的波动方程。

The project studies the long time behavior of solutions of nonlinear wave equations. It focuses on finding minimal hypotheses under which the solitary wave solutions of Schrödinger, Hartree, and Klein-Gordon equations are asymptotically stable. In addition, the work will investigate the decomposition of solutions into solitary and scattered waves (asymptotic completeness), as well as energy transfer among these waves. The research will not only develop new dispersive estimates for the semigroup of operators generated by the linearization around solitary waves whose shape is changing in time, but will also develop mathematical techniques to translate these estimates into results on the long time dynamics of the full, nonlinear equations.Wave equations are ubiquitous in quantum mechanics, statistical physics, optics, and acoustics, and they model certain phenomena in fluid dynamics, mechanics, and biology. Nonlinear wave equations support solitary waves, which are the building blocks for modeling such diverse phenomena as Bose-Einstein condensates, the propagation of information through optical fibers, and the formation of tsunamis. The detailed dynamics of these physical systems are still not fully understood. This project aims to create a mathematical theory for these phenomena and, more generally, for the long time propagation and interaction of solitary waves. It will focus on specific models from statistical physics and nonlinear optics, but the theory is expected to be sufficiently robust to apply to wave equations arising from models in other areas of science and engineering.