The Korteweg-de Vries Equation and Beyond

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The Korteweg-de Vries equation was originally introduced (in the 1890s) to give a simple effective explanation of the observation of solitary waves on the surface of shallow channels of water. Solitary waves travel large distances while maintaining their original shape. This is exceptional and represents a subtle interaction between two phenomena: (1) Dispersion, which results in longer waves traveling faster (e.g. tsunami travel much faster than the wind waves seen at the beach). (2) Nonlinearity, which makes large amplitude waves travel faster than shorter waves (this effect also manifests in waves traveling faster in deeper water). These two effects are important in many wave systems and controlling their interactions plays an important role also in technology, such as fibre-optic communication. The Korteweg--de Vries equation and its close relations provide a simple elegant framework both for deepening our understanding of these effects and also for educating future scientists, at all levels of preparation, about these important topics.The study of both invariant measures for dispersive PDE and stochastic forcing raise challenging questions in the low-regularity theory of dispersive PDE. A major theme of this project is to tackle such problems in the setting of the Korteweg-de Vries equation (KdV) by employing new techniques developed recently by M. Visan and the PI. In their current form, these methods exploit the complete integrability of KdV in a significant way. A key goal in selecting the progression of problems to be studied is to investigate how robust these methods are against perturbations that destroy this complete integrability. A second major component of the project is the adaptation of the new methods to other completely integrable systems, most notably the nonlinear Schrodinger equation (NLS) and the modified KdV equation (mKdV). The third facet of the project is the study of threshold solutions, namely, those that lie at the boundary between scattering and more complicated behaviour.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.