Solution concepts, well-posedness, and wave breaking for the Fornberg–Whitham equation

Abstract

We discuss concepts and review results about the Cauchy problem for the Fornberg–Whitham equation, which has also been called Burgers–Poisson equation in the literature. Our focus is on a comparison of various strong and weak solution concepts as well as on blow-up of strong solutions in the form of wave breaking. Along the way we add aspects regarding semiboundedness at blow-up, from semigroups of nonlinear operators to the Cauchy problem, and about continuous traveling waves as weak solutions.

Numerical Study on the Breaking Phenomena of the Fornberg-Whitham Equation

“…For more information on theoretical results known for the Whitham and the FW equations, see, for example, the monograph by Naumkin and Shishmarev [6], a recent review by Hörmann [5] and the references therein.…”

“…The time evolution of the discrete Fourier transform (DFT) ûk of u(x, t) is traced. Then, the computation of the integral term in (15), which represents the linear dispersion, can be performed by simply multiplying −iω k to ûk , where ω k is the frequency of the k-th mode given by the linear dispersion relation (5). The computation of the nonlinear term is performed by a pseudo-spectral way as shown schematically as follows:…”

“…Since B is the multiplicative coefficient of the dispersion term, an increase in B enhances the dispersive effect, making the occurrence of breaking less likely. The trend (ii) can also be understood roughly from the dispersion relation (5). The phase velocity c(k) and its derivative dc(k)/dk are given as Let us consider increasing b from a very small value while keeping B fixed, corresponding to moving vertically upward on the B-b plane as that shown in Fig.…”

Norm inflation and ill-posedness for the Fornberg-Whitham equation

“…Later, Haziot [15], Hörmann [19,21], Wei [26,27], Wu-Zhang [29] and Yang [30] sharpened this blowup criterion and presented the sufficient conditions about the initial data to lead the wave-breaking phenomena of the FW equation. The discontinuous traveling waves as weak solutions to the FW equation were investigated in [20].…”