{'fraction_non_alphanumeric': 0.013831258644536652, 'fraction_numerical': 0.0, 'mean_word_length': 5.766355140186916, '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}}
Nonlinear variational surface waves

We derive nonlocal asymptotic equations for weakly nonlinear surface wave solutions of variational wave equations in a half-space. These equations are analogous to, but different from, equations that describe weakly nonlinear Rayleigh waves in elasticity and other hyperbolic conservation laws. We prove short time existence of smooth solutions of a simplified, but representative, asymptotic equation and present numerical solutions which show the formation of cusp-singularities. This singularity formation on the boundary is a different mechanism for the nonlinear breakdown of smooth solutions of hyperbolic IBVPs from the more familiar one of singularity formation in the interior.