{'fraction_non_alphanumeric': 0.02599388379204893, 'fraction_numerical': 0.0, 'mean_word_length': 5.009174311926605, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Wave-breaking limit to the wake-field effect in an underdense plasma.

The influence of the sub-light-speed group velocity of a square-shaped laser pulse on a longitudinal wake-field wave is examined via a one-dimensional analysis of the electric potential equation. Analytical expressions for the potential, the electric field, the wake-field wavelength, and the optimum pulse width are obtained and fully agree with results from the numerical integration of the potential equation. The conditions for the onset of wave breaking due to sub-light-velocity pulses are given for both leading and trailing pulse edges, in agreement with particle simulations