Mathematical Sciences: Waves and Stability in Nonlinear Partial Differential Equations

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9403871 Pego Nonlinear waves play a significant role in systems of partial differential equations of importance in science and engineering. A primary goal of the proposed work is to develop effective methods for understanding the stability properties of many such waves. The proposer and collaborators have recently introduced some promising new methods for studying the stability of simplified models of solitary water waves. We plan to extend these methods to study waves in various systems of importance, including the full water wave equations and simplified models of particle physics and phase transitions. A second major topic of investigation will be the study of low velocity flow in fluids near the liquid-gas critical point, both in zero-G and 1-G conditions. High compressibility and anomalous physical properties characterize this regime, and several phenomena have been observed in experiments in spacecraft and on earth which are unexplained or only partly explained. The proposed stability analyses involve the use and further development of several mathematical tools, including the Evans function, and Krein's perturbation determinant. Both are analytic functions whose zeros correspond to eigenvalues or resonance poles. The use of linear stability analysis to prove nonlinear stability will be further developed for solitary waves of dispersive equations. Also, a new way of reformulating Hamiltonian systems, so that constraints become conserved quantities, will be investigated for: water waves, flexure of cables in zero-G, and anisotropic materials that are `stiff' in one direction. Recent developments in applied mathematics regarding `zero-Mach number flows' in combustion will be adapted and extended for studying near-critical pure fluids.