On nonlinear superposition of the KdV-Burgers shock waves and the behavior of solitons in a layered medium

Superposition of explicit (analytic) monotone non-increasing shock waves for the KdV-Burgers equation is studied. Initial profile chosen as a sum of two such shock waves gradually transforms into a single shock wave of a somewhat complex yet qualitatively predictable structure. The layered media consist of layers with both dispersion and dissipation and layers with dispersion but without dissipation. In the latter case the waves are described by the KdV equation, while in the former - by the Kdv-Burgers one. A soliton solution of the KdV equation meeting a layer with dissipation transforms somewhat similarly to a ray of light in air crossing semi-transparent plate. Both nonlinear superposition and the behavior of solitons are demonstrated in detail, modelled numerically and graphically presented.