Long nonlinear waves in a compressible magnetically structured atmosphere

The Hohlov-Zabolotskaja equation with an additional boundary condition is shown to describe long nonlinear small-amplitude fast sausage surface waves in a magnetic slab embedded in magnetic environment. It is proved that the obtained boundary problem has no solutions in the form of solitary waves. Approximate solution in the form of nonlinear stationary wave is found with the use of expansion in the power series of small amplitude. Second harmonic generation by a sinusoidal wave is studied. The law of energy conservation is obtained. Results of numerical computations are presented. They show that a sinusoidal disturbance does not overturn. The possibility of transmission of wave energy into corona along a magnetic slab is discussed in connection with these results.