{'fraction_non_alphanumeric': 0.0223463687150838, 'fraction_numerical': 0.0, 'mean_word_length': 5.107954545454546, '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 Integrable Nature of Modulational Instability

We investigate the nonlinear stage of the modulational (or Benjamin--Feir) instability by characterizing the initial value problem for the focusing nonlinear Schrodinger (NLS) equation with nonzero boundary conditions (NZBC) at infinity. We do so using the recently formulated inverse scattering transform (IST) for this problem. While the linearization of the NLS equation ceases to be valid when the perturbations have grown sufficiently large compared to the background, the results of the IST remain valid for all times and therefore provide a convenient way to study the nonlinear stage of the modulational instability. We begin by studying the spectral problem for the Dirac operator (i.e., the first half of the Lax pair for the NLS equation) with piecewise constant initial conditions which are a generalization to NZBC of a potential well and a potential barrier. Since the scattering data uniquely determine the time evolution of the initial condition via the inverse problem, the study of these kinds of poten...