We study theoretically and observe experimentally the evolution of periodic wave trains by utilizing surface gravity water wave packets. Our experimental system enables us to observe both the amplitude and the phase of these wave packets. For low steepness waves, the propagation dynamics is in the linear regime, and these waves unfold a Talbot carpet. By increasing the steepness of the waves and the corresponding nonlinear response, the waves follow the Akhmediev breather solution, where the higher frequency periodic patterns at the fractional Talbot distance disappear. Further increase in the wave steepness leads to deviations from the Akhmediev breather solution and to asymmetric breaking of the wave function. Unlike the periodic revival that occurs in the linear regime, here the wave crests exhibit self acceleration, followed by self deceleration at half the Talbot distance, thus completing a smooth transition of the periodic pulse train by half a period. Such phenomena can be theoretically modeled by using the Dysthe equation.