Existence of dipoles of Klein-Gordon-Zakharov system

TL;DR

This paper investigates the long-term behavior of solutions to the Klein-Gordon-Zakharov system, demonstrating the existence of special dipole solutions with solitary waves whose positions diverge logarithmically over time.

Contribution

The study introduces the existence of dipole solutions characterized by specific asymptotic behavior, using spectral analysis and coercivity estimates for the associated Hamiltonian.

Findings

Existence of solutions approaching a sum of solitary waves with diverging positions.

Positions of solitary waves grow logarithmically with time.

Method involves spectral analysis and compactness arguments.

Abstract

In this paper, we study the long time behavior of solutions of Klein-Gordon-Zakharov system. We show that there exists a solution with special characteristics, which we shall refer to as a dipole solution, that is, there exists a solution $\vec{u}$ such that $$\left\|\vec{u}(t)-\sum_{k=1}^{2}\vec{R}_{k}\right\|_{X} \to 0 \, \, \text{as}\, \, t\to \infty,$$ where $\vec{R}_{k}$ represents a solitary wave for each $k$, with a translation $z_k$ with respect to its position, satisfying that $$|z_1(t)-z_2(t)| \sim 2\log(t)\, \, \text{as} \, \, t\to \infty.$$ Our approach will initially focus on the spectral analysis of the Hamiltonian operator associated with our system. Subsequently, we aim to establish a coercivity estimate that will allow us to derive conditions ensuring the existence of our solution. It is important to note that, in this problem, our objective is to obtain approximate solutions by solving a final data problem. These approximate solutions will then be used, through uniform estimates and compactness results, to derive the desired conclusions via density arguments.