Global bifurcation of solitary waves for the Whitham equation

Abstract

The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnström and Wahlén. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve.

Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

“…For all c > 1, there exist smooth and rapidly decaying solitary wave solutions to the Whitham-Green-Naghdi system (WGN) with velocity c and such that the following holds. This is in sharp contrast to the celebrated result [6] on the existence of (peaked) solitary waves of extreme height for the water waves problem, and the corresponding result obtained on the Whitham equation [30,77] (see also [52] and references therein for a numerical investigation), and invalidates the naive thinking that this feature relies only on the dispersion relation of the equations linearized about the rest state.…”

“…The price to pay is that the equations include non-local pseudodifferential operators (Fourier multipliers). Whitham's prediction turned out to be valid at least for the unidirectional model which bears his name, as shown by [44,30,77,70]. This fact triggered renewed activity on bidirectional models (systems), and we refer to the surveys [51,15,22] for more information.…”

Solitary waves in a Whitham equation with small surface tension

“…24, 25 recently derived a generalization of the classical center-manifold theory that is applicable to a wide class of nonlocal problems, and this was further extended in Ref. 26 to an even wider class of nonlocal problems which, as we will show, includes (1). With this in mind, the primary goal of this paper is to use a nonlocal version of the center-manifold theorem and a corresponding normal-form reduction to establish the existence of small amplitude generalized solitary and modulated solitary-wave solutions to the gravity-capillary Whitham equation (1) in the small surface tension case.…”

“…In this work, we utilize instead an approach based on the recent nonlocal center-manifold reduction technique developed by Faye and Scheel 24,25 and further refined by Truong et al 26 As we will see, this set of techniques provides a unified approach for proving existence of both periodic and solitary waves for (6). The nonlocal center-manifold theorem bears resemblance to its classical local counterpart, that there exists a neighborhood in a uniform locally Sobolev space where the nonlocal equation is equivalent to a local finite-dimensional system of ODEs.…”

“…However, this framework does not fit nonlocal equations from hydrodynamics. To remedy this, Truong et al 26 have extended this result to a larger class of nonlocal equations. They also demonstrate the strength of this reduction technique and exemplify how to extract qualitative information on the solutions from the reduced ODE, which they use to construct an extreme solitary wave for the gravity Whitham equation.…”