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

Abstract For surface gravity waves, it is known that wave breaking may occur in the temporal evolution as a result ofthe 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 \(-\infty\).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 \(u_0(x)=\cos x\).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 regionson the parameter plane, is qualitatively explained in terms of the competition between the effects of dispersion and nonlinearity.