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Journal article
Asymptotic Stability of High-dimensional Zakharov-Kuznetsov Solitons
Archive for rational mechanics and analysis, v 220(2), pp 639-710
01 May 2016
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Abstract
We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Kortewegde Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrodinger (NLS) dynamics, are strongly asymptotically stable in the energy space. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard [Proc R Soc Edinburgh 126: 89-112, 1996]. Our proofs follow the ideas of Martel [SIAM J Math Anal 38: 759-781, 2006] and Martel and Merle [Math Ann 341: 391-427, 2008], applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition which is numerically tested for the two and three dimensional cases with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.
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Details
- Title
- Asymptotic Stability of High-dimensional Zakharov-Kuznetsov Solitons
- Creators
- Raphael Cote - École PolytechniqueClaudio Munoz - University of Paris-SudDidier Pilod - Federal University of Rio de JaneiroGideon Simpson - Drexel University
- Publication Details
- Archive for rational mechanics and analysis, v 220(2), pp 639-710
- Publisher
- Springer Nature
- Number of pages
- 72
- Grant note
- 291214 / ERC; European Research Council (ERC); European Commission DMS-1409018 / NSF; National Science Foundation (NSF) NC130017 / Millennium Nucleus Center for Analysis of PDE 1150202 / FONDECYT; Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT); CONICYT FONDECYT Fondo Basal CMM-Chile; Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT); CONICYT PIA/BASAL 1409018 / Division Of Mathematical Sciences; National Science Foundation (NSF); NSF - Directorate for Mathematical & Physical Sciences (MPS) 302632/2013-1; 481715/2012-6 / CNPq/Brazil; Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPQ)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000379423900005
- Scopus ID
- 2-s2.0-84958932789
- Other Identifier
- 991019168791404721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics, Applied
- Mechanics