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

For surface gravity waves, it is known that wave breaking may occur in the temporal evolution as a result of the steepening of waveform due to nonlinearity. Here, "breaking"refers to the phenomenon in which the slope of the front face of the wave diverges to -infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\infty $$\end{document}. The Fornberg-Whitham equation is a model equation which can reproduce this breaking phenomenon. In this study, the breaking phenomenon of the Fornberg-Whitham equation is investigated numerically. The equation is normalized to a form that includes two free parameters, while the initial condition is fixed as u0(x)=cosx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0(x)=\cos x$$\end{document}. The results are categorized in terms of whether wave breaking occurs or not in the course of the temporal evolution, and summarized as a scatter plot on the parameter plane. The overall shape of the critical curve, which separates the breaking and the non-breaking regions on the parameter plane, is qualitatively explained in terms of the competition between the effects of dispersion and nonlinearity.