Nonlinear waves in the solar atmosphere

In this paper, we give a brief review of the contemporary theory of nonlinear waves in the solar atmosphere. The choice of topics reflects personal interests of the author. Historically the theory of nonlinear waves was first applied to the solar atmosphere to explain the chromospheric and coronal heating. It was assumed that the turbulent motion in the solar convective zone excites sound waves that propagate upwards. Due to nonlinearity these waves steepen and form shocks. The wave energy dissipates in these shocks thus heating the corona. We give a brief description of propagation and damping of nonlinear sound waves in the stratified solar atmosphere, and point out that, at present, the acoustic heating remains the most popular theory of heating the lower chromosphere. Then we extend the analysis to nonlinear slow magnetosonic waves in coronal plumes and loops, and discuss its implications for interpretation of observational results. The next topic of interest is the propagation of nonlinear waves in a magnetically structured atmosphere. Here, we restrict our analysis to slow sausage waves in magnetic tubes and discuss properties of solitary waves described by the Leibovich-Roberts equation. We conclude with the discussion of nonlinear theory of slow resonant layers, and its possible application to helioseismology.