PhysRevE.86.036605.pdf (1.17 MB)
Download fileUndular bore theory for the Gardner equation
journal contribution
posted on 2015-03-16, 12:27 authored by A.M. Kamchatnov, Y.-H. Kuo, Tai-Chia Lin, T.-L. Horng, S.-C. Gou, Richard Clift, Gennady El, Roger GrimshawWe 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.
History
School
- Science
Department
- Mathematical Sciences
Published in
PHYSICAL REVIEW EVolume
86Issue
3Pages
? - ? (23)Citation
KAMCHATNOV, A.M. ... et al, 2012. Undular bore theory for the Gardner equation. Physical Review E, 86 (3), paper 036605, 23pp.Publisher
© American Physical SocietyVersion
- VoR (Version of Record)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Publication date
2012Notes
This paper was originally published in the journal Physical Review E (© American Physical Society) at: http://dx.doi.org/10.1103/PhysRevE.86.036605ISSN
1539-3755Publisher version
Language
- en