Focusing of nonlinear eccentric waves in astrophysical discs

We develop a fully nonlinear approximation to the short-wavelength limit of eccentric waves in astrophysical discs, based on the averaged Lagrangian method of Whitham (1965). In this limit there is a separation of scales between the rapidly varying eccentric wave and the background disc. Despite having small eccentricities, such rapidly varying waves can be highly nonlinear, potentially approaching orbital intersection, and this can result in strong pressure gradients in the disc. We derive conditions for the steepening of nonlinearity and eccentricity as the waves propagate in a radially structured disc in this short-wavelength limit and show that the behaviour of the solution can be bounded by the behaviour of the WKB solution to the linearised equations.