On the near soliton dynamics for the 2D cubic Zakharov-Kuznetsov equations

TL;DR

This paper analyzes the long-term behavior of solutions to the 2D cubic Zakharov-Kuznetsov equations near solitons, classifying possible outcomes and establishing conditions for blow-up or convergence.

Contribution

It provides a complete description of the asymptotic behaviors for solutions near solitons, including blow-up, finite-time departure, or convergence, using refined modulation and energy methods.

Findings

Solutions can blow up, leave the soliton tube, or converge to a soliton.

Near a non-positive energy soliton with above-threshold mass, solutions blow up.

The proof adapts techniques from gKdV equations and overcomes coercivity issues via a transform.

Abstract

In this article, we consider the Cauchy problem for the cubic (mass-critical) Zakharov-Kuznetsov equations in dimension two: $$\partial_t u+\partial_{x_1}(\Delta u+u^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^{2}.$$ For initial data in $H^1$ close to the soliton with 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; 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 mass-critical generalized Korteweg-de Vries equation (gKdV) equation in a similar context by Martel-Merle-Rapha\"el. 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 of the Schr\"odinger operator which appears in the virial-type estimate. To overcome the difficulty, we apply a transform, which was first introduced in Kenig-Martel [13], to perform the virial computations after converting the original problem to the adjoint one. Th coercivity of the Schr\"odinger operator in the adjoint problem has been numerically verified by Farah-Holmer-Roudenko-Yang [9].