This article presents a new method for analyzing arterial blood pressure waves. The technique is based on the scattering transform and consists in solving the spectral problem associated to a one-dimensional Schrödinger operator with a potential depending linearly upon the pressure. This potential is then expressed with the discrete spectrum which includes negative eigenvalues and corresponds to the interacting components of an N-soliton. The approach is analogous to the Fourier transform where the solitons play the role of sinus and cosinus components. The proposed method seems to have interesting clinical applications. It can be used for example to separate the fast and slow parts of the blood pressure that correspond to the systolic (pulse transit time) and diastolic phases (low velocity flow) respectively.