Existence and nonexistence of traveling waves of coupled Burgers’ equations

Traveling waves of a system of viscous coupled Burgers’ equations are classified in this paper. Traveling wave solutions of the scalar Burgers’ equation are simple, exhibiting a step-down or step-up wave pattern. Conversely, numerous different wave patterns can appear for the coupled Burgers’ system; these wave patterns are determined by the strength of the coupling constants and other parameters. Comprehension of the solutions of these traveling waves needs to precede the study of various other aspects of the system, such as the stability of numerical schemes. We point out that those rich interacting patterns observed in this study provide an important class of special solutions that deserve tests over various numerical schemes, in particular to suppress spurious oscillations that have been reported in the literature. It turns out that eight different parameter regimes account for the entire system with relevant parameters. For each of the eight regimes, we completely characterize the existence and nonexistence of traveling waves within a class that we introduced in the study. We observed left-moving and right-moving waves involving a variety of wave patterns. Waves of crossing patterns, where one species steps up and the other steps down, and bump-like patterns were shown to exist. We also provided numerical results for a selected set of traveling waves to illustrate the established results of existence and nonexistence.