Travelling-wave analysis and identification. A scattering theory framework

This article presents a new travelling waves analysis and identification method based on scattering theory. This inverse scattering technique consists in solving the spectral problem associated to a one-dimensional Schrodinger operator perturbed by a potential depending upon the wave to analyze, and optimized in order to approximate this wave by an isospectral flow in the sense of Lax. In this method, the interacting components of an N-soliton are the elementary travelling waves for the approximation. These N solitons play an analogous role to linear superpositions of sinus and cosinus in the Fourier analysis of standing waves. In the proposed analysis of travelling waves, low and high frequency components are replaced by low and high velocity components. Two applications of the method are presented. The first one concerns the identification of an N-soliton and is illustrated with N = 3. The second one consists in the analysis of the arterial blood pressure waves during the systolic phase (pulse transit time) and the diastolic phase (low velocity flow).