Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons

TL;DR

This paper proves the strong asymptotic stability of Zakharov-Kuznetsov solitons in high dimensions, extending known stability results and introducing new analytical tools like a Liouville property and a Virial identity.

Contribution

It extends stability analysis of ZK solitons to high dimensions, introducing new monotonicity properties, a Liouville theorem, and a Virial identity without additional spectral assumptions.

Findings

Proved strong asymptotic stability of ZK solitons in energy space.

Established a new Liouville type property characterizing ZK solitons.

Developed a Virial identity applicable to high-dimensional ZK dynamics.

Abstract

We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schr\"odinger (NLS) dynamics, are strongly asymptotically stable in the energy space in the physical region. 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. Our proofs follow the ideas by Martel and Martel and Merle, 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.