Asymptotic K-soliton-like Solutions of the Zakharov-Kuznetsov type equations

Abstract

We study here the Zakharov-Kuznetsov equation in dimension $2$ and $3$ and the modified Zakharov-Kuznetsov equation in dimension $2$. Those equations admit solitons, characterized by their velocity and their shift. Given the parameters of $K$ solitons $R^k$ (with distinct velocities), we prove the existence and uniqueness of a multi-soliton $u$ such that ${\Vert u-{\sum }_{k=1}^K{R}^k\Vert }_{H^1}\to 0$ as $t\to +\infty$. The convergence takes place in $H^s$ with an exponential rate for all $s\ge 0$. The construction is made by successive approximations of the multi-soliton. We use classical arguments to control of $H^1$-norms of the errors (inspired by Martel [21]), and introduce a new ingredient for the control of the $H^s$-norm in dimension $d\ge 2$, by a technique close to monotonicity.