Stability in nonlinear wave equations

该项目研究了非线性波动方程解的长时间行为。它致力于寻找薛定谔方程、Hartree方程和Klein-Gordon方程的孤立波解渐近稳定的极小假设。此外，这项工作将研究解分解为孤立波和散射波(渐近完备性)，以及这些波之间的能量转移。非线性波动方程支持孤波，孤波是模拟各种现象的基础，如玻色-爱因斯坦凝聚、通过光纤的信息传播和海啸的形成。这些物理系统的详细动力学仍未完全了解。这个项目的目的是为这些现象，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。它将重点放在统计物理和非线性光学的特定模型上，但该理论预计将足够健壮，足以应用于其他科学和工程领域的模型所产生的波动方程。

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.