Abstract
We consider the critical generalized Zakharov–Kuznetsov (ZK) equation, \(u_t + \partial _{x_1}(\Delta u + u^3) = 0, (x_1,x_2) \in {\mathbb {R}}^2\). In Farah et al. (Instability of solitons in the 2d cubic Zakharov–Kuznetsov equation, arXiv:1711.05907, 2017), we proved that solitons are unstable for this equation following the strategy by Martel and Merle (GAFA Geom Funct Anal 11:74–123, 2001) in their study of the critical generalized Kortweg–de Vries equation. The main ingredient used in Farah et al. (Instability of solitons in the 2d cubic Zakharov–Kuznetsov equation, arXiv:1711.05907, 2017) was the new pointwise decay estimates in two dimensions together with monotonicity properties of solutions. In this paper, we show that using only monotonicity properties and not relying on pointwise estimates, thus, greatly simplifying the approach, we can prove an instability of solitons, though a slightly weaker version.
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Acknowledgements
L.G.F. was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES, Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq and Fundação de Amparo a Pesquisa do Estado de Minas Gerais - Fapemig/Brazil. Since September 2017, J.H. has been serving as Program Director in the Division of Mathematical Sciences at the National Science Foundation (NSF), USA, and as a component of this job, J.H. received support from NSF for research, which included work on this paper. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. S.R. was partially supported by the NSF CAREER Grant DMS-1151618 and NSF Grant DMS-1815873.
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Farah, L.G., Holmer, J. & Roudenko, S. On instability of solitons in the 2d cubic Zakharov–Kuznetsov equation. São Paulo J. Math. Sci. 13, 435–446 (2019). https://doi.org/10.1007/s40863-019-00142-7
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DOI: https://doi.org/10.1007/s40863-019-00142-7