{'fraction_non_alphanumeric': 0.02050580997949419, 'fraction_numerical': 0.0, 'mean_word_length': 5.565022421524664, '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 Evolution Equations for Parallel Electromagnetic Multispecies Plasma Modes: Reductive Perturbation Theory in the Linear Wave Frame and Electric Field Generation

A variant of the reductive perturbation theory is introduced, describing nonlinear phenomena in a frame moving with the linear phase velocity. The stretching is simpler and less terms are to be carried in expanding the basic equations to different orders, but the method only works well when the displacement current can be neglected, for plasmas that are not too strongly magnetized. Within this framework, the derivative nonlinear Schrödinger (DNLS) equation for parallel propagating electromagnetic waves has been (re)derived in ordinary multi-ion plasmas, with due attention to the stage at which a parallel electric field is generated and charge neutrality no longer holds. A simple reasoning shows that the DNLS equation cannot have stationary solitary waves, only envelope solitons. For critical plasma compositions the coefficient of the dispersive term in the DNLS vanishes, necessitating a different stretching and leading to a vector modified Korteweg-de Vries (VmKdV) equation. The latter is encountered not only in electron-positron plasmas, but other plasmas at critical densities are possible, e.g. dusty plasmas. Although the VmKdV equation is nonintegrable, in contrast to the DNLS equation, it can have stationary solitary solutions with linearly polarized magnetic fields.