On weakly nonlinear waves in media exhibiting mixed nonlinearity. (Q1410269)

The authors obtain the transport equations governing small amplitude high frequency disturbances, that include both quadratic and cubic nonlinearities inherent in hyperbolic systems of conservation laws. They derive the coefficients of the nonlinear terms in the transport equation in terms of the Glimm interaction coefficients. For symmetric and isotropic systems the mean curvature of the wave front, which appears as the coefficient of the linear term in the transport equation, is shown to be related to the derivative of the ray tube area along the bicharacteristics. They demonstrate that the amplitude of the disturbance should become unbounded in the neighbourhood of the point where the ray tube collapses. In addition, they derive a formula, akin to the one obtained by \textit{R. Rosales} [An introduction to weakly nonlinear geometric optics. Proc. IMA Workshop 1989, IMA Vol. Math. Appl. 29, 281--310 (1991; Zbl 0725.76083)], for the energy dissipated across shocks.