Spatial chaos in weakly dispersive and viscous media: a nonperturbative theory of the driven KdV-Burgers equation

Abstract

The asymptotic travelling wave solution of the KdV-Burgers equation driven by the long scale periodic driver is constructed. The solution represents a shock-train in which the quasi-periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions. It is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes. Two types of chaotic behavior are considered. The first type is a weak chaos, where only a small chaotic deviation from the periodic solution occurs. The second type corresponds to the developed chaos where the solution ``ignores'' the driver period and represents a random sequence of uncorrelated shocks. In the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor, while in the case of developed chaos it moves on a repellor. Both solutions depend on a parameter indicating the reference shock position in the shock-train. The structure of a one dimensional set to which this parameter belongs is investigated. This set contains measure one intervals around the fixed points in the case of periodic or weakly chaotic solutions and it becomes a fractal in the case of strong chaos. The capacity dimension of this set is calculated.