On the Near Soliton Dynamics for the 2D Cubic Zakharov–Kuznetsov Equations

Abstract

In this article, we consider the Cauchy problem for the cubic (mass-critical) Zakharov-Kuznetsov equations in dimension two:

$$\begin{aligned} \partial _tu+\partial _{x_1}(\Delta u+u^3)=0,\quad (t,x)\in [0,\infty )\times {\mathbb {R}}^{2}. \end{aligned}$$

For the initial data in \(H^{1}\) close to the soliton and satisfying a suitable space-decay property, we fully describe the asymptotic behavior of the corresponding solution. More precisely, for such initial data, we show that only three possible behaviors can occur: (1) The solution leaves a tube near soliton in finite time; (2) the solution blows up in finite time; and (3) the solution is global and locally converges to a soliton. In addition, we show that for initial data near a soliton with non-positive energy and above the threshold mass, the corresponding solution will blow up as described in Case 2. Our proof is inspired by the techniques developed for the mass-critical generalized Korteweg de Vries (gKdV) equation in a similar context by Martel-Merle-Raphaël [36]. More precisely, our proof relies on refined modulation estimates and a modified energy-virial Lyapunov functional. The primary challenge in our problem is the lack of coercivity for the Schrödinger operator, which appears in the virial-type estimate. To overcome the difficulty, we apply a transform, which was first introduced in Kenig-Martel [14], to perform the virial computations after converting the original problem into an adjoint one. The coercivity of the Schrödinger operator in the adjoint problem has been numerically verified by Farah-Holmer-Roudenko-Yang [9].