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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
68/1998

On transitions to stationary states in 1D nonlinear wave equations

Alexander Komech

Abstract

We develop the theory of attractors for finite energy solutions to conservative nonlinear wave equations in a whole space. For "nondegenerate" equations the attractor coincides with the set of all finite energy stationary states. The convergence to the attractor holds as $t \to \pm \infty$ in the Fréchet topology defined by local energy seminorms. The proof of the attraction is based on the investigation of energy scattering to infinity. The results give a mathematical model of N.Bohr's transitions to quantum stationary states ("quantum jumps").

Received:
23.01.99
Published:
23.01.99
MSC Codes:
35L70, 37K40, 37K45
Keywords:
attractor, stationary state, fréchet topology, energy scattering to infinity, goursat problem

Related publications

inJournal
1999 Repository Open Access
Alexander Komech

On transitions to stationary states in one-dimensional nonlinear wave equations

In: Archive for rational mechanics and analysis, 149 (1999) 3, pp. 213-228