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Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

Published online by Cambridge University Press:  08 August 2019

Naoki Sato Open the ORCID record for Naoki Sato [Opens in a new window]  and
Michio Yamada
Naoki Sato*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Michio Yamada
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
*
†Email address for correspondence: sato@kurims.kyoto-u.ac.jp
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Abstract

The problem of linear instability of a nonlinear travelling wave in a canonical Hamiltonian system with translational symmetry subject to superharmonic perturbations is discussed. It is shown that exchange of stability occurs when energy is stationary as a function of wave speed. This generalizes a result proved by Saffman (J. Fluid Mech., vol. 159, 1985, pp. 169–174) for travelling wave solutions exhibiting a wave profile with reflectional symmetry. The present argument remains true for any non-canonical Hamiltonian system that can be cast in Darboux form, i.e. a canonical Hamiltonian form on a submanifold defined by constraints, such as a two-dimensional surface wave on a constant shearing flow, revealing a general feature of Hamiltonian dynamics.

JFM classification

Mathematical Foundations: Hamiltonian theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 876 , 10 October 2019 , pp. 896 - 911
Copyright
© 2019 Cambridge University Press 

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