Applied Analysis for Integrable Nonlinear Waves

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Waves in nature (for example, water waves or electromagnetic waves) can be modeled by solutions of certain differential equations that incorporate various physical processes into a mathematical framework that admits further detailed study in principle. But even with such a wave equation in hand, there remains the difficult task of deducing important information from the model, information that is needed to solve important problems of engineering that motivate the study of waves in the first place. One common approach is to use computers to solve the equations (approximately). However, such an approach is limited in scope to very concrete simulations involving particular initial conditions, and it holds only in parameter ranges in which the numerical methods can be accurate. On the other hand, there are other parameter regimes that are quite common (for example, the situation that electromagnetic waves can be approximated by light rays) in which the computer-based approach becomes difficult, and therefore one would like to have an alternative method of analysis. This project is aimed at developing such alternative methods of asymptotic analysis for wave propagation problems that are nonlinear (so that large waves can be accurately modeled) but that nonetheless admit a kind of transform relating them to linear problems (for which the familiar superposition principle applies). One application of such theoretical analysis would be to describe the evolution of sub-surface oil plumes caused by an oil leak like the Deepwater Horizon disaster. This project is an attempt to place completely integrable nonlinear wave equations on a similar footing as constant-coefficient linear equations, from the point of view of asymptotic analysis (i.e., to further develop nonlinear analogues of the classical methods of stationary phase and steepest descent for integrals). The specific problems to be addressed include the study of the inverse-scattering transform for the Benjamin-Ono equation (a nonlocal integrable model for internal gravity waves) in the small-dispersion limit, the study of the resonant interaction of wave packets through a quadratic nonlinearity in the semiclassical limit, the study of an analogue of the defocusing nonlinear Schroedinger equation describing waves in two spatial dimensions in the semiclassical limit, the study of dynamical stability of so-called rogue waves, the study of mixed initial/boundary-value problems for integrable equations in the semiclassical limit, and the investigation of "universal wave patterns", analogues to wave propagation problems of universal phase transitions in statistical mechanics and mathematical physics.