Mkn 421 was repeatedly observed with BeppoSAX in 1997-1998. This is the first of two papers where we present the results of a thorough temporal and spectral analysis of all the data available to us, focusing in particular on the flare of April 1998, which was simultaneously observed also at TeV energies. Here we focus on the time analysis, while the spectral analysis and physical interpretation are presented in the companion paper. The detailed study of the flare in different energy bands reveals very important new results: i) hard photons lag the soft ones by 2-3 ks -a behavior opposite to what is normally found in high energy peak BL Lacs X-ray spectra; ii) the flare light curve is symmetric in the softest X-ray band, while it becomes increasingly asymmetric at higher energies, with the decay being progressively slower than the rise; iii) the flux decay of the flare can be intrinsically achromatic if a stationary underlying emission component is present. The temporal and spectral information obtained challenge the simplest models currently adopted for the (synchrotron) emission and most importantly provide clues on the particle acceleration process.
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