Dynamics of the collision of two nearly equal solitary waves for the Zakharov-Kuznetsov equation

TL;DR

This work extends Martel–Merle's two-soliton collision framework to the Zakharov-Kuznetsov equation in dimensions 2 and 3, where solitary waves are non-explicit. The authors construct an intrinsic approximate two-soliton solution and perform a refined modulation analysis, aided by a modified energy functional and transverse-parameter estimates, to show that solutions near a two-soliton configuration decompose into two modulated solitary waves plus a small $H^1$-error for all times. A detailed bootstrap argument around a distance function $Z(t)$, governed by a Hamiltonian-type ODE, controls the collision dynamics and guarantees stability across the collision, followed by asymptotic stability results. The methods combine intrinsic interaction corrections, nonlocal modulation, and Hamiltonian structure, offering a robust toolkit for studying high-dimensional, non-explicit solitary waves in focusing dispersive equations.

Abstract

We study the dynamics of the collision of two solitary waves for the Zakharov-Kuznetsov equation in dimension $2$ and $3$. We describe the evolution of the solution behaving as a sum of $2$-solitary waves of nearly equal speeds at time $t=-\infty$ up to time $t=+\infty$. We show that this solution behaves as the sum of two modulated solitary waves and an error term which is small in $H^1$ for all time $t \in \mathbb R$. Finally, we also prove the stability of this solution for large times around the collision. The proofs are a non-trivial extension of the ones of Martel and Merle for the quartic generalized Korteweg-de Vries equation to higher dimensions. First, despite the non-explicit nature of the solitary wave, we construct an approximate solution in an intrinsic way by canceling the error to the equation only in the natural directions of scaling and translation. Then, to control the difference between a solution and the approximate solution, we use a modified energy functional and a refined modulation estimate in the transverse variable. Moreover, we rely on the hamiltonian structure of the ODE governing the distance between the waves, which cannot be approximated by explicit solutions, to close the bootstrap estimates on the parameters. We hope that the techniques introduced here are robust and will prove useful in studying the collision phenomena for other focusing non-linear dispersive equations with non-explicit solitary waves.

Figures (4)

• Figure 1: Graph of the functions $Z$ and $\dot{Z}$
• Figure 2: Schematic reperesentation of the two solitary waves $R_1$ and $R_2$, of the functions $W_1$ and $W_2$, and of the function $P$ with a plateau between the solitary waves
• Figure 3: Phase portrait of equation \ref{['defi:eq_Z_general']}. In red: the constant solution. In blue: the four solutions defined on $\mathbb{R}$ ending at $(0,0)$ when $t$ goes to $-\infty$ or $+\infty$. In black: all the other trajectories.
• Figure 4: Comparison of a trajectory $(z,\dot{z})$ in red with the phase portrait associated to the equations related to $H_-$ (left), $H$ (middle) and $H_+$ (right). On each graph, the three lines are associated to three values $h_0<h<h_1$.

Theorems & Definitions (137)

• Proposition 1.1
• Remark 1.2
• Proposition 1.3
• Remark 1.4
• Theorem 1.5: Dynamics of the interaction of two solitary waves
• Remark 1.6
• Theorem 1.7: Stability in the region of collision
• Remark 1.8
• Remark 1.9
• Proposition 2.1