A quantitative description of the flagellar dynamics in the procyclic T. brucei is presented in terms of stationary oscillations and traveling waves. By using digital video microscopy to quantify the kinematics of trypanosome flagellar waveforms. A theoretical model is build starting from a Bernoulli-Euler flexural-torsional model of an elastic string with internal distribution of force and torque. The dynamics is internally driven by the action of the molecular motors along the string, which is proportional to the local shift and consequently to the local curvature. The model equation is a nonlinear partial differential wave equation of order four, containing nonlinear terms specific to the Korteweg-de Vries (KdV) equation and the modified-KdV equation. For different ranges of parameters we obtained kink-like solitons, breather solitons, and a new class of solutions constructed by smoothly piece-wise connected conic functions arcs (e.g. ellipse). The predicted amplitude and wavelengths are in good match with experiments. We also present the hypotheses for a step-wise kinematical model of swimming of procyclic African trypanosome.
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