Abstract
We consider the two dimensional generalization of the Korteweg-de Vries (KdV) equation, the generalized Zakharov-Kuznetsov (ZK) equation \(u_t + \partial _{x_1}(\Delta u + u^p) = 0, (x_1,x_2) \in \mathbb R^2\). It is known that solitons are stable for nonlinearities p < 3 and unstable for p > 3, which was established by de Bouard (Proc R Soc Edinb Sect A 126:89–112, 1996) generalizing the arguments of Bona et al. (Proc R Soc Lond 411:395–412, 1987) for the gKdV equation. The L 2-critical case with p = 3 has been open and in this paper we prove that solitons are unstable in the cubic ZK equation. This matches the situation with the critical gKdV equation, proved in 2001 by Martel and Merle (Geom Funct Anal 11:74–123, 2001). While the general strategy follows (Martel and Merle, Geom Funct Anal 11:74–123, 2001), the two dimensional case creates several difficulties and to deal with them, we design a new virial-type quantity, revisit monotonicity properties and, most importantly, develop new pointwise decay estimates, which can be useful in other contexts.
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Notes
- 1.
The nonlinearity in such gZK equation should be understood as \(\partial _{x_1}(|u|{ }^{p-1} u)\).
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Acknowledgements
Most of this work was done when the first author was visiting GWU in 2016–2017 under the support of the Brazilian National Council for Scientific and Technological Development (CNPq/Brazil), for which all authors are very grateful as it boosted the energy into the research project. S.R. would like to thank IHES and the organizers for the excellent working conditions during the trimester program “Nonlinear Waves” in May–July 2016. L.G.F. was partially supported by CNPq, CAPES and FAPEMIG/Brazil. J.H. was partially supported by the NSF grant DMS-1500106. S.R. was partially supported by the NSF CAREER grant DMS-1151618/1929029.
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Farah, L.G., Holmer, J., Roudenko, S. (2019). Instability of Solitons in the 2d Cubic Zakharov-Kuznetsov Equation. In: Miller, P., Perry, P., Saut, JC., Sulem, C. (eds) Nonlinear Dispersive Partial Differential Equations and Inverse Scattering. Fields Institute Communications, vol 83. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9806-7_6
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