Viscous Profiles of Traveling Waves in Scalar Balance Laws: The Canard Case

Abstract

The traveling wave problem for a viscous conservation law with a nonlinear source term leads to a singularly perturbed problem which necessarily involves a non-hyperbolic point. The correponding slow-fast system indicates the existence of canard solutions which follow both stable and unstable parts of the slow manifold. In the present paper we show that for the viscous equation there exist such heteroclinic waves of canard type. Moreover, we determine their wave speed up to first order in thesmall viscosity parameter by a Melnikov-like calculation after a blow-up near the non-hyperbolic point. It is also shown that there are discontinuous waves of the inviscid equation which do not have a counterpart in the viscous case.