{'fraction_non_alphanumeric': 0.011235955056179775, 'fraction_numerical': 0.0, 'mean_word_length': 4.6879432624113475, '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}}
The head-on interaction of two solitary waves of unequal amplitude

In this paper we consider the interaction of two unequal solitary waves travelling in opposite directions. The interaction equations are derived from the perturbation expansion of the Euler equations, which lead to the Boussinesq equation at lowest order. These equations are recast to obtain two weakly coupled KdV equations which are solved by the method of inverse scattering. We show that the amplitude of each solitary wave after interaction is reduced. This change in the amplitudes is shown to be fifth order in ε, the order of magnitude of the amplitude of the waves. This is in agreement with the change of amplitude produced by the interaction of two equal waves which arises in the reflection of a wave by a vertical wall.