Slow Nonlinear Waves in Magnetic Flux Tubes

A theory of weakly nonlinear slow waves in magnetic flux tubes is developed in the ideal MHD approximation. Fairly simple approximate dispersion relations are derived that are valid for waves of arbitrary wavelength. These dispersion relations make it possible to obtain a number of new model evolutionary equations for body and surface slow waves in magnetic flux tubes. It is established that there are two families of exact analytic solutions to the equations for weakly nonlinear slow waves. It is found that both the body and surface solitary waves can be in the form of either contractions or bulges running along the tube. A model Korteweg-de Vries-Burgers equation is derived and generalized to waves of arbitrary wavelength. It is shown that exact analytic solutions to these equations correspond to shock waves and hydraulic jumps (or bores) with nonoscillating fronts