Discontinuous travelling wave solutions for certain hyperbolic systems

In an earlier paper on a malignant cell invasion model (Marchant et al., SIAM J. Appl. Math, 60, 2000) we introduced a novel form of discontinuous travelling wave solution. These solutions could be studied easily by combining behaviour within a phase plane with the Rankine-Hugoniot shock conditions, which describe properties (such as the ratio of the jump discontinuities to the speed of propagation) that solutions may possess. These results were new for several reasons. The shock conditions relate to hyperbolic equations (which the model is) but were applied in a travelling wave ordinary differential equation phase plane using techniques that usually apply to parabolic reaction-diffusion systems. In addition the solutions possess singular behaviour near several points in the phase plane but in spite of this there exists a robust and stable family of physically interesting solutions. In this paper we discuss two previously studied models, one of detonation theory and one of angiogenesis. We show that each of these models also possesses a family of discontinuous travelling wave solutions which was not previously discovered. Of particular interest is the solution which has a blunt interface at the front of the invading profile. In all three models it is this solution that is seen to stably evolve from physically relevant initial data, and for physically relevant parameter values. This work confirms the robustness of these novel travelling wave solutions and their applicability to a wider range of mathematical modelling situations.