The Structure Function Method for Asymmetric Solitary Waves

We present a method for the reconstruction of the waveform of asymmetric solitary waves which are observed in weakly dispersive, nonlinear media. Starting from the morphology of the waveform, we first give local differential laws which govern the waveform about its extrema and at infinity. Then we analytically extend these laws to find a differential equation of the generalized Korteweg-de Vries family for the waveform, holding over the whole real axis. A multi-valued, singular functional dependence of the nonlinear term of this equation on the amplitude of the solitarywave results, whichwas not considered before and which is intimately connected with the asymmetric nature of the solitarywave. If cast in terms of a suitable combination of thewave amplitude— a combination which we call the structure function of the solitary wave—the differential equation reduces to a modified Korteweg-de Vries equation with a polynomial nonlinear term, which we solve by quadrature, including up to sixth-order nonlinearities. In particular, the fourth-order accurate waveforms are given in terms of elementary functions, and they are in excellent agreement with observations. For a given maximum wave amplitude, several waveforms are found, each having a different width. This width decreases as the maximum wave amplitude increases.