Waves, structures, and the Riemann problem for a system of hyperbolic conservation laws

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.