Derivation of mean-field equations and dynamics of coherent structures in nonlinear dispersive PDE

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This project deals with certain types of equations modeling wave propagation in various physical contexts. Among these are (1) the structure of a collection of ultra-cold atoms (2) light waves moving through a medium whose index of refraction depends upon the strength of the wave allowing for the self-focusing of the wave (3) water waves in a shallow channel, and (4) waves that move through the electrons and ions in a plasma. Among the important mathematical issues are the rigorous derivation of such models from fundamental physical laws, and the precise study of special types of coherent wave solutions (variously referred to as solitary waves, kinks, vortices, and monopoles, and blow-up profiles). In the last decade, new techniques from several branches of mathematics, specifically harmonic analysis, spectral theory, and Hamiltonian mechanics, have led to striking developments and the resolution of many problems once deemed intractable. The PI has outlined several conjectures that he and his students and collaborators will address by adapting and extending these new methods. At the same time, the resolution of these conjectures is expected to extend the applicability of these methods and interface other branches of mathematics.First, several conjectures on the derivation of the nonlinear Schrodinger equation (NLS) as a limit of the fundamental quantum mechanical model for a many-body system of bosons are outlined. The PI will continue to investigate problems pertaining to dimensional reduction by the imposition of strong anisotropic confining, and allowing attractive interactions leading to the focusing NLS, which supports bright solitary wave solutions, as observed in recent experiments. Second, problems pertaining to the geometrical structure of singularity formation for NLS are proposed. In particular, the PI will consider the stability of new contracting sphere blow-up solutions, the possibility for blow-up on an ellipse, and a precise description of blow-up in the Zakharov system, a model for collapse of Langmuir waves in a plasma. Third, the PI proposes to consider the dynamics of dark solitons, vortices, and monopoles, which are solutions for which the wave function has nontrivial topological structure on the spatial boundary, in equations such as the sine Gordon equation, subjected to semiclassical perturbations. Fourth, the dynamics of non-perturbative interacting multi-solitons in completely integrable models, specifically the modified Korteweg-de Vries (mKdV) and NLS equations, subjected to semiclassical perturbation, will also be pursued.