Nonlinear Dispersive Hamiltonian Systems: Solitary Waves and Global Attractors

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Nonlinear Dispersive Hamiltonian Systems: Solitary Wavesand Global Attractors.Abstract of Proposed ResearchAndrew Comech This project is to study the stability of solitary wave solutions of nonlinear dispersive Hamiltonian systems. He is particularly interested in analyzing the instability of the critical solitons in the Korteweg- de Vries equation and the stability of discrete peakons and breathers. Another topic is the orbital stability of solitary waves in the nonlinear Dirac equations. Tools to be used here include dispersive estimates and normal form analysis. He also plans to investigate the theory of attractors in infinite dimensional Hamiltonian systems and, in particular, to determine when such systems have finite dimensional attractors. The equations to be studied appear in ocean dynamics, in the atmosphere and in quantum field theories. Their analysis will help understand associated physical phenomena on scales ranging from the very smallest electronics and chips to large weather patterns.