ATTRACTORS AND TRANSIENTS FOR A PERTURBED PERIODIC KDV EQUATION - A NONLINEAR SPECTRAL-ANALYSIS

In this paper we rigorously show the existence and smoothness in epsil on of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameter epsilon. The shape and the spectr al transforms of these traveling waves are calculated perturbatively t o first order. A linear stability theory using squared eigenfunction b ases related to the spectral theory of the KdV equation is proposed an d carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of t he solutions.