Stability criterion for solitons of the ZK-type equations
E.A. Kuznetsov
TL;DR
This paper generalizes linear stability criteria for solitons from KDV-type equations to ZK-type equations, deriving a Vakhitov-Kolokolov stability condition applicable to various nonlinearities, highlighting the link to Hamiltonian unboundedness and collapse.
Contribution
It extends stability analysis to ZK-type solitons and derives a universal Vakhitov-Kolokolov criterion for arbitrary nonlinearities.
Findings
Stability criterion derived for ZK-type solitons.
Instability linked to Hamiltonian unboundedness.
Collapse associated with power nonlinearity instability.
Abstract
Early results concerning the linear stability of the solitons in equation of the KDV-type \cite{KUZNETSOV1984314} are generalized to solitons describing by the ZK-type equation. The linear stability criterion for ground solitons in the Vakhitov-Kolokolov form is derived for such equations with arbitrary nonlinearity. For the power nonlinearity the instability criterion coincides with the condition of the Hamiltonian unboundedness from below. The latter represents the main feature for appearance of collapse in such systems.
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