Well-posedness and Long-time Behavior of Dispersive Integrable Systems

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Integrable systems have long served as guides in the study of Hamiltonian partial differential equations. They arise as effective models of real physical systems, including in optics and many-body quantum mechanics. It is in the setting of completely integrable systems that solitons and multisolitons were first discovered. These structures have since found numerous applications in the applied sciences: for example, in fiber optics, solitons have been employed in the transmission of digital signals over long distances, while in biology, they are used to describe signal propagation in the nervous system and low-frequency collective motion in proteins. This project seeks to investigate both longstanding and newly introduced integrable models. Specifically, we seek to find the minimal conditions on the initial state under which one can construct global-in-time dynamics, investigate the (in)stability of special structures (such as solitons and multisolitons), and elucidate the long-time behavior of general solutions. The project provides significant research training opportunities for graduate students, who are integrated into the main objectives of the project. The project investigates the following specific questions for the newly introduced continuum Calogero-Moser equations: (1) large data well-posedness in the scaling-invariant space, (2) scattering for both the defocusing model and the focusing equation for initial data with mass less than that of the ground state soliton, and (3) the determination of the blowup threshold in the focusing case. Further objectives include orbital and asymptotic stability of multisoliton solutions to the Benjamin-Ono equation in optimal well-posedness spaces, dispersive decay away from the soliton component for large solutions to this equation, and the construction of Gibbs dynamics for the Landau-Lifshitz model. 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.