Infinite-Dimensional Dynamical Systems - Stability and Long-Time Behavior

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A variety of phenomena arising in nature and applications, such as ocean waves (rogue waves) and wave breaking, optical transmission lines and optical communications, novel materials (graphene) among others, involve single-wave structures (solitons) that travel without change of shape. This project is aimed at the study of the nonlinear partial differential equations, which are instrumental for analysis and modeling of such soliton-like structures. Many important phenomena most readily manifest themselves through the behavior of the special solutions such as traveling or standing waves, which not only serve as basis for the behavior of the system, but also determine the related nearby dynamics. These structures and their stability are of great importance and are essential in practical applications. Stable states of the system attract all nearby configurations, while the loss of stability or being unable to control the dynamics is of practical importance as well. The investigator will involve undergraduate and graduate students in various stages of the project and will aim to recruit and retain them to continue working in the field of applied mathematics. Exposing students to parts of the project that require broad interaction with other sciences such as optics, water waves and liquid crystals, will be particularly beneficial for the students' training.Throughout the project, the point of view in working with these partial differential equations will be one of infinite-dimensional dynamical systems, which allows us to take advantage of the classical tools by adapting them to the infinite-dimensional setting. The project focuses on Hamiltonian models with sign indefinite energy functionals such as various Dirac systems. The goal is to investigate the linear and spectral stability for certain solitary waves, but also to prove results on uniform bounds for the spectrally stable solutions. Investigating the dynamics near solitary waves for some exotic NLS models and for water wave models such as the Benney-Luke equations is another focus of this proposal. The study of the long-term dynamics and asymptotic profiles in the Landau - de Gennes models of liquid crystals rounds up this research program. All these directions will require new techniques and tools from diverse areas such as functional analysis, dynamical systems, and harmonic analysis as well as numerical simulations. The local and long-time behavior of solutions, as well as their stability is of great practical importance as they describe systems in optics, liquid crystals, and water waves, among others.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.