Applications of the inverse spectral transform to a Korteweg-de Vries equation with a Kuramoto-Sivashinsky-type perturbation.

Abstract

In this dissertation, the initial-boundary value problem u(t) - uuₓ + δ²uₓₓₓ + uₓₓ + β²uₓₓₓₓ = 0. u(x + 1) = u(x); u(x,0) = u(I)(x) is studied analytically and numerically. This partial differential equation is a hybrid between the well-known Korteweg-deVries and Kuramoto-Sivashinsky equations. It is shown numerically that this problem has for strong dispersion (δ² ≫ 1) travelling wave attractors which can be constructed as perturbations of cnoidal waves of the Korteweg-deVries equation. The perturbation theory is extended to the spectral structure and the linear stability of these travelling waves. The linear stability theory makes use of the squared eigenfunction basis related to the spectral theory of the Korteweg-deVries equation. This yields better estimates of the linear stability than those previously known. This seems to be the first use of the squared eigenfunction basis in the study of dissipative perturbations of the Korteweg-deVries equation. Next, the equations of motion for the action and angle variables of the KdV-equation are written down for the perturbed flow and the transient and attracting phases of the dynamics of the initial-boundary value problem are interpreted with these equations. A numerical study of the dynamics of these 'spectral coordinates' exhibits a series of interesting phenomena. In certain parameter regions a mode reduction is considered and a perturbation theory of the action and angle variables is applied to the truncated system. Finally, the effects of an additional uniform damping term νu in the initial-boundary value problem are discussed. We also compiled various ideas and concepts for an analytical proof of the existence of travelling wave attractors for strong dispersion. They might serve as guidelines for the actual proof which is still missing. A theoretical appendix presents some proofs and calculations to complement the main text and a numerical appendix describes the computational setup in the numerical study of the initial-boundary value problem.