Hugoniot Locus of Coupled Burgers’ Equations and (non-)Existence of Traveling Viscous Shocks

Traveling waves in the system of viscous coupled Burgers' equations in one spatial dimension exhibit rich patterns, depending on the signs and magnitudes of densities and the coeficents. This contrasts with the well-known fact in the scalar equation, where the discontinuities of admissible nonnegative shocks are always downward jumps. For the two-species system, we characterize three types of patterns: $(i)$ Fractioned waves where one species increases while the other decreases; $(ii)$ bi-sigmoid waves where both species either increase or decrease; $(iii)$ waves with a bump. We explore the relationship between viscous shocks and Hugoniot Locus, following a classical approach. We specify the Hugoniot Locus and prove (non-)existence of viscous shocks for elements satisfying certain admissibility conditions. Interestingly, fractioned waves are exclusively viscous 1-shocks. 2-shocks are bi-sigmoid waves, and waves with a bump are over-compressive shocks. We present numerical results for the traveling waves shown to exist in our main theorem.