Gino Biondini, State University of New York at Buffalo
601 Pao Yue-Kong Library
Fifty years after Zabusky and Kruskal’s discovery of solitons, there still remain many fundamental open questions about nonlinear waves. This talk is devoted to two classical problems involving singular asymptotic limits: (i) the nonlinear stage of modulational instability and (ii) the small dispersion limit of (2+1)-dimensional systems.
(i) Modulational instability (MI), namely the instability of a constant background to long-wavelength perturbations, is a ubiquitous nonlinear phenomenon discovered in the 1960’s. However, a characterization of the nonlinear stage of MI – namely, the behavior of solutions once the perturbations have become comparable with the background – was missing. In the first part of the talk I will describe recent work on this subject. I will first show how MI manifests itself in the inverse scattering transform for the focusing nonlinear Schrodinger (NLS) equation. Then I will characterize the nonlinear stage of MI by computing the long-time asymptotics of solutions of the NLS equation for localized perturbations of a constant background. For long times, the space-time plane divides into three regions: a left far field and a right far field, in which the solution is approximately constant, and a central region in which the solution is described by a slowly modulated traveling wave. Finally, I will show that this kind of behavior is not limited to the NLS equation, but instead it is shared by many different nonlinear models (including several PDEs, nonlocal systems and differential-difference equations).
(ii) In the 1960s, G.B. Whitham formulated a method that allows one to study the small-dispersion limit of a nonlinear PDE by deriving a set of hyperbolic PDEs describing the modulation of the parameters of the traveling-wave solutions of original system. Whitham modulation theory, as is now called, has been subsequently generalized and applied with great success in a variety of settings. Most results, however, are limited to PDEs in one spatial dimension. In the second part of the talk I will show how one can formulate a (2+1)-dimensional generalization of Whitham modulation theory to derive the genus-1 Whitham modulation equations for a number of systems, including the Kadomtsev-Petviashvili (KP) equation, the two-dimensional Benjamin-Ono equation and a modified KP equation. I will discuss some basic properties of the resulting Whitham systems and I will show how these systems can be used to investigate many interesting questions about solutions of the original PDE, including the temporal dynamics of certain initial conditions and the transverse stability of genus-1 solutions.