Abstract

In many applications it is natural to observe “locally periodic” patterns. Such patterns appear spatially periodic on local space/time scales while their fundamental wave characteristics (such as amplitude or frequency) may slowly change, i.e., modulate, over large space/time scales. One powerful tool used to study such structures is Whitham’s theory of wave modulations, commonly referred to as Whitham theory. While Whitham theory lacks rigorous justification in general, its predictions match remarkably well with physical and numerical observations, and it has been rigorously justified for a growing number of models and equations. In the recent work by Binswanger et al., the authors use Whitham modulation theory to analyze a generalized Whitham equation (i.e., a Whitham-type equation with generalized nonlinear flux and linear dispersion relation) and establish a modulational instability criterion. Building on their work, this talk will rigorously connect the modulational instability criterion from the work by Binswanger et al. to the spectral stability of long-wavelength perturbations of periodic traveling wave solutions to the generalized Whitham equation, thereby justifying Whitham modulation theory for the generalized Whitham equation. This provides a succinct justification of Whitham modulation theory for the various equations that can be written in the form of the generalized Whitham equation.