English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK)equation, a physically relevant high dimensional generalization of theKorteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixedKdV and nonlinear Schr\"odinger (NLS) dynamics, are strongly asymptoticallystable in the energy space in the physical region. We also prove that the sumof well-arranged solitons is stable in the same space. Orbital stability of ZKsolitons is well-known since the work of de Bouard. Our proofs follow the ideasby Martel and Martel and Merle, applied for generalized KdV equations in onedimension. In particular, we extend to the high dimensional case severalmonotonicity properties for suitable half-portions of mass and energy; we alsoprove a new Liouville type property that characterizes ZK solitons, and a keyVirial identity for the linear and nonlinear part of the ZK dynamics, obtainedindependently of the mixed KdV-NLS dynamics. This last Virial identity relieson a simple sign condition, which is numerically tested for the two and threedimensional cases, with no additional spectral assumptions required. Possibleextensions to higher dimensions and different nonlinearities could be obtainedafter a suitable local well-posedness theory in the energy space, and theverification of a corresponding sign condition.

Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons