Mathematical Sciences: Scattering and Stability of NonlinearWaves

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Work on this mathematics project is grouped into three main areas: (a) nonlinear scattering and orbital asymptotic stability in infinite dimensional Hamiltonian systems, (b) transitions to instability and (c) vector Zakharov and nonlinear Schrodinger equations. The first class of problems is concerned with the long-time behavior of solutions of nonlinear dispersive systems in terms of a nonlinear bound state channel (modeling nondecaying behavior) and a dispersive radiation field. The governing equations have the structure of a finite dimensional dynamical system which is coupled to the infinite dimensional dispersive radiation field. In the second, a new theory for a class of eigenvalue problems will be applied to nonlinear equations modeling long wave propagation in dispersive media. An advanced formulation is being developed which is independent of the dynamical systems framework. Applications to nonlocal equations arising in the study of fluids and plasmas will be sought. The third goal is to study the existence and properties of nonlinear bound states of the vector Zakharov and Schrodinger equations. These systems are a better approximation of the physics of waves in a collisionless plasma. Associated with these equations are believed to be phenomena not present in scalar systems, such as non-isotropic ground states, which may participate in the dynamics of singularity formation. Differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.