RUI: Orbital and Asymptotic Stability of Solutions to Peakon Equations

Many natural phenomena, including water waves, atmospheric flows, and other complex systems, are governed by nonlinear partial differential equations whose solutions can develop features such as sharp wave crests, wave breaking, and coherent structures. Understanding whether these structures persist under perturbations is a fundamental challenge with implications for prediction, modeling, and simulation across the physical sciences. This project investigates the mathematical stability of a class of nonlinear wave equations known as "peakon equations", which were originally derived to model shallow water waves and exhibit wave forms with sharp, cusp-like peaks. By developing new analytical and computational tools to understand the long-term behavior of these waves, the project will advance knowledge in nonlinear dynamics and applied mathematics. The research will also provide training opportunities for undergraduate and graduate students in mathematical analysis, scientific computing, and computer-assisted proof techniques, helping prepare a workforce with interdisciplinary skills increasingly important in science, engineering, and technology. Through publications, conference presentations, and educational activities, the project will contribute to the broader mathematical community. This project studies the orbital and asymptotic stability of solutions to several families of nonlinear partial differential equations admitting peakon solutions, including the Camassa--Holm, Degasperis--Procesi, Novikov, and related b-family equations. Despite substantial progress in recent years, many fundamental questions remain unsolved because of challenges arising from non-smooth solutions, nonlocal operators, and conserved quantities that are only well defined in restricted function spaces. The project will develop new analytical techniques, numerical methods, and computer-assisted proofs to establish existence and stability results for wave solutions and to resolve open problems suggested by numerical observations. Particular attention will be given to explaining stability transitions and bifurcation phenomena observed in parameterized families of equations, including discrepancies between existing theory and computational evidence. Numerical computations and computer-assisted proofs will be used both to discover new dynamical behaviors and to provide rigorous verification of various stability results. The mathematical methods developed in this work are expected to contribute broadly to the study of nonlinear wave propagation, dynamical systems, and computational mathematics, while providing tools that may be applicable to other problems involving complex nonlinear phenomena. 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.