| Issue |
Math. Model. Nat. Phenom.
Volume 13, Number 2, 2018
Shock Waves, Discontinuities and Singularities in Natural Phenomena
|
|
|---|---|---|
| Article Number | 23 | |
| Number of page(s) | 20 | |
| DOI | https://doi.org/10.1051/mmnp/2018024 | |
| Published online | 22 June 2018 | |
Normal form for transverse instability of the line soliton with a nearly critical speed of propagation★
1
Department of Mathematics, McMaster University,
Ontario,
Hamilton
L8S 4K1, Canada
2
Department of Applied Mathematics, Nizhny Novgorod State Technical University,
24 Minin Street,
603950
Nizhny Novgorod, Russia
* Corresponding author: dmpeli@math.mcmaster.ca
Received:
9
June
2017
Accepted:
10
October
2017
In the context of the line solitons in the Zakharov–Kuznetsov (ZK) equation, there exists a critical speed of propagation such that small transversely periodic perturbations are unstable if the soliton speed is larger than the critical speed and orbitally stable if the soliton speed is smaller than the critical speed. The normal form for transverse instability of the line soliton with a nearly critical speed of propagation is derived by means of symplectic projections and near-identity transformations. Justification of this normal form is provided with the energy method. The normal form predicts a transformation of the unstable line solitons with larger-than-critical speeds to the orbitally stable transversely modulated solitary waves.
Mathematics Subject Classification: 35Q53 / 37K40 / 37L10
Key words: Zakharov-Kuznetsov equation / line soliton / transverse instability / normal form
© EDP Sciences, 2018
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