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Asymptotic stability of a finite sum of solitary waves for the Zakharov-Kuznetsov equation

Didier Pilod, Frédéric Valet

TL;DR

The paper analyzes the Zakharov-Kuznetsov equation in dimensions $d=2$ and $d=3$ and proves the asymptotic stability of a finite sum of well-ordered solitary waves, extending the Martel–Merle–Tsai framework to higher dimensions. The authors combine modulation theory, energy and mass conservation, almost-monotonicity on oblique half-spaces, and nonlinear Liouville (rigidity) results around solitary waves (CMPS16 for 2D, FHRY23 for 3D) to handle the interaction of two solitons and derive precise long-time behavior. A quantitative orbital stability result is established with explicit bounds depending on the initial velocity gap and separation, and the speeds are shown to converge to limiting values while the solution converges to a sum of two moving ground states on a rightward region. These results provide a rigorous foundation for multi-soliton dynamics and collision analysis for ZK in higher dimensions and illustrate how higher-dimensional geometry can be controlled in a non-integrable dispersive framework.

Abstract

We prove the asymptotic stability of a finite sum of well-ordered solitary waves for the Zakharov-Kuznetsov equation in dimensions two and three. Moreover, we derive a qualitative version of the orbital stability result which turns out to be useful for the study of the collision of two solitary waves. The proof extends the ideas of Martel, Merle and Tsai for the sub-critical gKdV equation in dimension one to the higher-dimensional case. It relies on monotonicity properties on oblique half-spaces and rigidity properties around one solitary wave introduced by Côte, Muñoz, Pilod and Simpson in dimension two, and by Farah, Holmer, Roudenko and Yang in dimension three.

Asymptotic stability of a finite sum of solitary waves for the Zakharov-Kuznetsov equation

TL;DR

The paper analyzes the Zakharov-Kuznetsov equation in dimensions and and proves the asymptotic stability of a finite sum of well-ordered solitary waves, extending the Martel–Merle–Tsai framework to higher dimensions. The authors combine modulation theory, energy and mass conservation, almost-monotonicity on oblique half-spaces, and nonlinear Liouville (rigidity) results around solitary waves (CMPS16 for 2D, FHRY23 for 3D) to handle the interaction of two solitons and derive precise long-time behavior. A quantitative orbital stability result is established with explicit bounds depending on the initial velocity gap and separation, and the speeds are shown to converge to limiting values while the solution converges to a sum of two moving ground states on a rightward region. These results provide a rigorous foundation for multi-soliton dynamics and collision analysis for ZK in higher dimensions and illustrate how higher-dimensional geometry can be controlled in a non-integrable dispersive framework.

Abstract

We prove the asymptotic stability of a finite sum of well-ordered solitary waves for the Zakharov-Kuznetsov equation in dimensions two and three. Moreover, we derive a qualitative version of the orbital stability result which turns out to be useful for the study of the collision of two solitary waves. The proof extends the ideas of Martel, Merle and Tsai for the sub-critical gKdV equation in dimension one to the higher-dimensional case. It relies on monotonicity properties on oblique half-spaces and rigidity properties around one solitary wave introduced by Côte, Muñoz, Pilod and Simpson in dimension two, and by Farah, Holmer, Roudenko and Yang in dimension three.
Paper Structure (20 sections, 37 theorems, 209 equations, 7 figures)

This paper contains 20 sections, 37 theorems, 209 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Zakharov-Kuznetsov equation
  3. Solitary waves
  4. Finite sum of solitary waves
  5. Outline of the proof
  6. Notation
  7. Decomposition and Properties of a solution close to the sum of 2 solitons
  8. Properties of the linearized operator around the ground state
  9. Modulation close to the sum of decoupled solitary waves
  10. Evolution of the energy
  11. Almost monotonicity of the mass on the right
  12. Quantified orbital stability
  13. Bootstrap setting and proof of Theorem \ref{['theo:orbital']} (i)
  14. Proof of Proposition \ref{['propo:a_priori']}
  15. Asymptotic stability
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $d=2$ or $d=3$ and let $0<\underline{c}<\bar{c}$ be fixed. There exist positive constants $k=k(\underline{c},\bar{c})$, $K=K(\underline{c},\bar{c})$ and $A=A(\underline{c},\bar{c})$ such that the following is true. For two velocities $c_1^0$ and $c_2^0$ satisfying $\underline{c}<c_2^0<c_1^0<\bar Let $\alpha<\alpha^\star$, $Z> Z^\star$, $(z_i^0, \omega_i^0) \in \mathbb{R} \times \mathbb{R}^{d-1

Figures (7)

  • Figure 1: Scheme of the localization of the two solitary waves and the line $x=m(t)$ at two different times
  • Figure 2: Evolution of the position of the weight function from the time $t$ to $0$.
  • Figure 3: Evolution of the position of the line $\tilde{x}=0$ at two different times : $t=0$ on the left and $t=t_0>0$ on the right.
  • Figure 4: Evolution of the localization of the weighted mass on the left of the first solitary wave.
  • Figure 5: Evolution of the weight function at times $t_0$ and $T_1$
  • ...and 2 more figures

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • Proposition 2.2: Choice of modulation parameters
  • Remark 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5: Evolution of the modulation parameters
  • ...and 50 more