Undular bore theory for the Gardner equation

A. M. Kamchatnov, Y. H. Kuo, T. C. Lin, T. L. Horng, S. C. Gou, R. Clift, G. A. El, R. H.J. Grimshaw

Research output: Contribution to journalArticle

43 Citations (Scopus)

Abstract

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.

Original languageEnglish
Article number036605
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume86
Issue number3
DOIs
Publication statusPublished - 2012 Sep 18

Fingerprint

Modulation
modulation
cavities
Korteweg-de Vries Equation
Riemann Invariants
Rarefaction Wave
Phenomenology
phenomenology
Shock Waves
elastic waves
Invertible
Wave Propagation
shock waves
wave propagation
Discontinuity
mathematical models
discontinuity
nonlinearity
Composite
Nonlinearity

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics

Cite this

Kamchatnov, A. M., Kuo, Y. H., Lin, T. C., Horng, T. L., Gou, S. C., Clift, R., ... Grimshaw, R. H. J. (2012). Undular bore theory for the Gardner equation. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 86(3), [036605]. https://doi.org/10.1103/PhysRevE.86.036605
Kamchatnov, A. M. ; Kuo, Y. H. ; Lin, T. C. ; Horng, T. L. ; Gou, S. C. ; Clift, R. ; El, G. A. ; Grimshaw, R. H.J. / Undular bore theory for the Gardner equation. In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 2012 ; Vol. 86, No. 3.
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Kamchatnov, AM, Kuo, YH, Lin, TC, Horng, TL, Gou, SC, Clift, R, El, GA & Grimshaw, RHJ 2012, 'Undular bore theory for the Gardner equation', Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 86, no. 3, 036605. https://doi.org/10.1103/PhysRevE.86.036605

Undular bore theory for the Gardner equation. / Kamchatnov, A. M.; Kuo, Y. H.; Lin, T. C.; Horng, T. L.; Gou, S. C.; Clift, R.; El, G. A.; Grimshaw, R. H.J.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 86, No. 3, 036605, 18.09.2012.

Research output: Contribution to journalArticle

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AU - Lin, T. C.

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