Nonlinear modulation of spatially periodic waves
A CAMS Colloquium mathematical lecture with Kevin Zumbrun of Indiana University.
Abstract: Periodic waves are important features of solutions of nonlinear evolution systems in such varied contexts as optics, hydrodynamics, and reaction diffusion. A formal description of their behavior under perturbation is given by WKB expansion in terms of modulations in phase and local waveform, as pioneered by Whitham, Howard-Kopell, and Serre in various contexts. The Whitham modulation equations take the form, to lowest order, of a first-order system of conservation laws, whose characteristic speeds play a role in the nonlinear setting analogous to that of group velocity in the linear case, giving the rate of propagation of localized wave packets. In this talk we discuss recent results giving rigorous verification of this formal Whitham description using a combination of Bloch transform techniques, and techniques originating from shock wave stability and the theory of conservation laws for efficiently extracting nonlinear modulations in phase. Notably, this approach allows the treatment of situations for which the Whitham equations have multiple characteristic speeds, whereas previous techniques based on renormalization methods were limited to the case of a single characteristic speed. Indeed, the techniques introduced apply also in situations far from a periodic background, to which the Whitham equations no longer directly apply.