WEAKLY NONLINEAR WAVES FOR A CLASS OF LINEARLY UNSTABLE HYPERBOLIC CONSERVATION LAWS WITH SOURCE TERMS

摘要

We consider a pair of general hyperbolic conservation laws with source terms, and focus on the class of problems that are unstable in the linearized sense. We derive evolution equations governing the leading approximation of the nonlinear solution using multiple scale expansions. We then analyze these evolution equations to determine conditions under which linearly unstable disturbances equilibrate. In particular, we show that for certain parameter values periodic initial disturbances evolve into travelling waves consisting of piecewise continuous profiles joined by shocks. We also exhibit a novel bifurcation process whereby the wave number of the travelling wave increases a unit amount as a parameter value in the evolution equation is doubled. Numerical solutions are provided throughout. [References: 10]