The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity

Source abstract

Using the viscosity-capillarity admissibility criterion for shock waves, we solve the Riemann problem for the system of conservation laws where σ( u ) = u 3 − u . This system is hyperbolic at ( u , v ) unless . We find that the Riemann problem has a unique solution for all data in the hyperbolic regions, except for a range of data in the same phase (i.e. on the same side of the nonhyperbolic strip). In the nonunique cases, there are exactly two admissible solutions. The analysis is based upon a formula describing all saddle-to-saddle heteroclinic orbits for a family of cubic vector fields in the plane.