Asymptotic Stability of High-dimensional Zakharov-Kuznetsov Solitons

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.