Decay of coherent structures in damped Hamiltonian systems

G. Derks, E. van Groesen, T. Valkering

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Abstract

In numerical and physical experiments it is often observed that finite and infinite dimensional systems exhibit a low dimensional behaviour, in the sense that the dynamics looks as if it can be described with a few parameters. Often a spatially coherent structure is characteristic for the phenomenon. A well known example is a solitary wave, characterized by its phase and its amplitude. In this paper we consider a Hamiltonian system (or a Poisson system), that has an additional constant of motion (besides the Hamiltonian). We show coherent structures in such a system, by describing some solutions with 2 parameters, induced by the constant of motion. Further we demonstrate that the coherent structures survive even in cases where a small perturbation, such as dissipation, is present. This is demonstrated in some detail for a spherical pendulum with uniform friction, for the Korteweg-de Vries equation with uniform damping and for the Korteweg-de Vries-Burgers equation.
Original languageEnglish
Title of host publicationIntegration of Theory and Applications in Applied Mechanics
Subtitle of host publicationChoice of papers presented at the First National Mechanics Congress, April 2–4, 1990, Rolduc, Kerkrade, The Netherlands
EditorsJ.F. Dijksman, F.T.M. Nieuwstadt
Place of PublicationDordrecht
PublisherKluwer Academic Publishers
Pages329-340
Number of pages11
ISBN (Electronic)978-94-009-2125-2
ISBN (Print)978-94-010-7456-8
DOIs
Publication statusPublished - 1990
Event1st National Mechanics Congress 1990 - Rolduc, Kerkrade, Netherlands
Duration: 2 Apr 19904 Apr 1990
Conference number: 1

Conference

Conference1st National Mechanics Congress 1990
CountryNetherlands
CityRolduc, Kerkrade
Period2/04/904/04/90

Keywords

  • Solitary wave
  • Hamiltonian systems
  • Coherent structures
  • Relative equilibrium
  • Additional constant

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    Derks, G., van Groesen, E., & Valkering, T. (1990). Decay of coherent structures in damped Hamiltonian systems. In J. F. Dijksman, & F. T. M. Nieuwstadt (Eds.), Integration of Theory and Applications in Applied Mechanics: Choice of papers presented at the First National Mechanics Congress, April 2–4, 1990, Rolduc, Kerkrade, The Netherlands (pp. 329-340). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-009-2125-2_24