Existence and Energetic Stability of Traveling Waves in the Presence of Symmetry

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Traveling waves are solutions to time-dependent systems that propagate without changing their shape. They are observed throughout the natural world and include surface and internal waves in the ocean, ignition fronts in combustion theory, and even stripe patterns in animal fur. Through their ubiquity, traveling waves command the interest of researchers in nearly every corner of the physical and biological sciences. This project aims to develop novel tools for constructing large-amplitude waves, especially for the important case of pulses, which are localized disturbances that travel through unbounded domains (such as tsunami waves or rough waves). In a related direction, the project will apply and extend a new theoretical framework for diagnosing stability of traveling waves solutions, i.e., capacity of the waves to persist when subjected to a disturbance. This project will advance the mathematical understanding of traveling waves; the results will have implications to oceanography, fluid mechanics, and combustion theory. More generally, the techniques developed by this study are intended to provide a framework with considerable potential for future applications to a broad array of disciplines. Graduate students will be trained and actively involved in this research. This project aims to advance the mathematical understanding of traveling waves along two parallel tracks: existence theory and stability theory. The first set of activities concern the existence of large-amplitude traveling waves on unbounded domains. While many tools currently exist for constructing small-amplitude waves in a neighborhood of known explicit solutions, the non-perturbative regime is far less well understood. This is particularly true for problems set on unbounded domains, for which issues of compactness seriously frustrate what tools are available. This project will develop a global bifurcation theoretic machinery designed to overcome these obstructions using symmetry and monotonicity properties. This new framework will then be used to address important open problems in a variety of systems. Specifically, these applications include (i) construct large-amplitude bore solutions to a two-phase fluid system in a channel; (ii) prove the existence of large-amplitude traveling waves evolving according to general non-compact symmetry groups, for example scroll ring solutions to reaction-diffusion equations; and (iii) extend beyond the appearance of internal stagnation points a family of large-amplitude solitary stratified water waves constructed in earlier work. A second set of projects will develop new systematic tools for diagnosing the orbital stability or instability of traveling solutions to abstract Hamiltonian systems that possess symmetries. This work aims to relax key hypotheses in existing theory to allow more direct application to highly nonlinear systems like those governing water waves. Using this machinery, it is planned to (i) prove the instability of internal waves in a two-phase system confined to a channel and (ii) give a new systematic proof of the stability and instability of Korteweg?de Vries solitons. Moreover, a local well-posedness theory in the Hadamard sense for the water wave problem with a point vortex will be provided. 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.