Nonlinear waves in thin magnetic flux tubes in astrophysical plasma

A weakly nonlinear fundamental mode in a thin magnetic flux tube imbedded i n magnetized plasma is considered. It is revealed, that the fundamental mod e appears as a slow surface wave for C-T(2) > C-Ae(2) > and, as a slow body wave for C-T(2) < C-Te(2) . In the special case of C-Te(2) < C-T(2) < C-Ae (2), the fundamental mode became leaky due to emission of slow waves by the tube. An approximate dispersion relation for the fundamental mode is obtai ned. it makes possible to derive the generalized Leibovich-Roberts (GLR) eq uation, which governs the fundamental mode in a thin magnetic flux tube imb edded in a magnetized plasma. It is shown, that the KdV equation can be use d as a crude approximation of the GLR equation for long enough waves. It he lps to confirm, that the fundamental mode appears as humps of the tube for C-T(2) < C-Te(2), and, as a narrowing of the tribe for C-T(2) < C-Te(2). It is pointed out the shortcomings of the GLR equation doe to a false maximum of the dispersion, which appears in the approximate dispersion relation in comparison with the exact one. A nonlinear equation for the fundamental mo de in thin tubes, which does not have these shortcomings, is obtained.