Asymptotic $K$-soliton-like solutions of the Zakharov-Kuznetsov type equations

Abstract:

We study here the Zakharov-Kuznetsov equation in dimension $2$, $3$ and $4$ 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 \[ \| u(t) - \sum _{k=1}^K R^k(t) \|_{H^1} \to 0 \quad \text {as} \quad t \to +\infty . \] 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 [Amer. J. Math. 127 (2005), pp. 1103–1140]), and introduce a new ingredient for the control of the $H^s$-norm in dimension $d\geq 2$, by a technique close to monotonicity.