Nonlinear evolution equations, asymptotics and applications

The most familiar manifestation of nonlinear dispersive waves is perhaps that of a breaking ocean wave. However, nonlinear dispersive waves are ubiquitous in nature, appearing in many fields ranging from water waves to optics, acoustics, ferromagnetics, condensed matter, cosmology and beyond. When dissipation and nonlinearity are the dominant physical effects in a system, the regime of small dissipation often gives rise to shock waves (the most familiar manifestation being a sonic boom). The analogous phenomenon when dissipation is replaced by dispersion is that of dispersive shock waves. These are non-stationary coherent, multiscale oscillatory structures. Physical media giving rise to dispersive shock waves range from water waves to superfluids, nonlinear photonics, and magnetic spin systems. For example, in fluid dynamics, dispersive shocks are known as undular bores. Even though much work has been done over the past fifty years to understand these phenomena, many fundamental questions remain. A first component of this project involves the development of mathematical tools for studying the behavior of solutions of certain nonlinear evolution equations describing nonlinear wave phenomena in more than one spatial dimension. The second component of the project involves the application of these tools to study a variety of areas of interest, ranging from water waves to optics, networks, and statistical physics. Finally, the project will also serve as a vehicle for training several Ph.D. students. The mathematical study of nonlinear media giving rise to dispersive shocks often leads to certain systems of hyperbolic conservation laws. Over the last fifty years, various methods have been applied with success to study these kinds of systems. However, the behavior of solutions of dispersive nonlinear wave equations in more than one spatial dimension is not as well understood. In particular, a mathematical characterization of formation and propagation of dispersive shocks in two spatial dimensions is still largely an open problem. Recent work by the PI has opened up new avenues to study some long-standing open problems in this regard. Specifically, this project comprises four classes of problems: (i) Use of the Whitham modulation equations for the Kadomtsev-Petviashvili equation (the so-called KP-Whitham equations, which were recently derived by the PI) to study the temporal evolution of piecewise-constant soliton initial data and the formation and dynamics of dispersive shock waves in 2+1 dimensions. (ii) Study of the integrability structure of the above-mentioned KP-Whitham equations and their exact solutions. (iii) Derivation and application of Whitham modulation equations for various (2+1)-dimensional evolution equations of nonlinear Schrodinger type. (iv) Application of small dispersion limits and Whitham modulation theory to characterize phase state diagrams of p-star networks and certain random matrix models. The overarching theme of the project is to advance our understanding of dispersive wave phenomena in more than one spatial dimension. 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.