On the Existence of Periodic Traveling Waves to the Whitham Equation

Abstract

We study the existence of periodic traveling waves to the Whitham equation, which is a nonlinear, nonlocal and dispersive differential equation proposed by Whitham as a model for surface gravity waves featuring the exact linear dispersion relation for water waves. Based on a work by Ehrnström et al. we consider a generalized Whitham equation with power nonlinearities n(u) = |u|^q or u|u|^(q - 1) for q in (1, 5). It is found that there exist periodic traveling waves for all sufficiently large periods in any Sobolev space H^s of order s in (1/2, q), or s in (1/2, infinity) if n(u) = u^q and q = 2, 3 or 4. The waves are shown to be of small amplitude, measured by the H^s norm, for a subset of the orders. In addition, we provide an explicit lower bound on the wave speeds.

The existence technique treats the Whitham equation as the Euler-Lagrange equation of a constrained minimization problem. As a background we perform a detailed study of Fourier series and Sobolev spaces with arbitrary periods and the calculus of variations.