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

A system of hyperbolic conservation laws $$\partial_t u + \partial_x \partial_u Q = 0, \quad Q = u_1^3 / 3 + u_1 u_2^2, \qquad u = u(x,t) \in\mR^2,$$ as well as its viscous regularization $$\partial_t u + \partial_x \partial_u Q = \calM \partial_x^2 u, \qquad \calM = \diag (\mu_1,\mu_2), \quad \mu_1>0,\, \mu_2>0,$$ 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 $\pm\infty$, the solution also contains undercompressive shocks and Jouguet waves.