A system of hyperbolic conservation laws ₜ u + ₓ ᵤ Q = 0, Q = u₁³ / 3 + u₁ u₂², u = u (x, t) ², as well as its viscous regularization ₜ u + ₓ ᵤ Q = ₓ² u, = (μ₁, μ₂), μ₁>0, \, μ₂>0, are studied. It is assumed that admissible shocks are those that satisfy the requirement of existence of a structure (the traveling wave criterion). A solution of the Riemann problem is constructed that consists of rarefaction waves and shocks with structure. Depending on the conditions imposed at, the solution also contains undercompressive shocks and Jouguet waves.
Чугайнова et al. (Thu,) studied this question.