Summary: A generalization of the Korteweg-de Vries equation (KdV) is considered in which additional terms belonging to the Kuramoto-Sivashinsky equation (KS) are incorporated to account for a production-dissipation (input-output) energy balance. Two different situations are thoroughly investigated.
First, the production-dissipation part of the equation is taken as a small perturbation to the KdV, proportional to a smallness parameter . It is shown that within times limited by (and beyond) the KdV seches interact similarly to the Zabusky and Kruskal’s findings, save the “aging” they experience. At longer times the localized solutions adopt the terminal shape and phase velocity, and different humps can form bound states. The increase of the production-dissipation parameter exaggerates the effects through reducing the “practical infinity” for the time scale. For 2 with the rest of parameters equal to unity, the solution goes chaotic. These results outline the region where the long enough transients can be approximately considered as solitons, albeit imperfect ones.
The second situation is when the KS part of the equation is predominant. This happens either when is not small enough or for very long times (t or t) when strictly permanent shapes are attained which are in fact short waves and the dissipation (higher-order derivative) is dominant. It is shown that the solution to KS of homoclinic shape (a hump) does not qualify for a wave-particle/soliton since it does not persist as a permanent shape after collision and yields to a chaotic regime. The heteroclinic shapes (kinks/bores/hydraulic jumps/shocks) do behave as particles but the interactions appear to be completely inelastic. After two such wave-particles collide they stick to each other and deform to produce a single structure of the same, kind which carries the total momentum of the system. This kind of (really imperfect) solitons may be called “clayons” to emphasize the fact that upon collisions they behave as clay balls.
Thus the Zabusky and Kruskal’s soliton concept is extended in two directions: to “long” transients practically “permanent” and solitonic in the time scale set by production-dissipation processes and to true permanent wave-particles with, however, inelastic behaviour upon collisions.