Wave trains, solitons and modulation theory in FPU chains

Abstract

We present an overview of recent results concerning wave trains, solitons and their modulation in FPU chains. We take a thermodynamic perspective and use hyperbolic scaling of particle index and time in order to pass to a macroscopic continuum limit. While strong convergence yields the well-known p-system of mass and momentum conservation, we generally obtain a weak form of it in terms of Young measures. The modulation approach accounts for microscopic oscillations, which we interpret as temperature, causing convergence only in a weak, average sense. We present the arising Whitham modulation equations in a thermodynamic form, as well as analytic and numerical tools for the resolution of the modulated wave trains. As a prototype for the occurrence of temperature from oscillation-free initial data, we discuss various Riemann problems, and the arising dispersive shock fans, which replace Lax-shocks. We predict scaling and jump conditions assuming a generic soliton at the shock front.