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

Abstract

We study here the Zakharov-Kuznetsov equation in dimension 2 2 , 3 3 and 4 4 and the modified Zakharov-Kuznetsov equation in dimension 2 2 . Those equations admit solitons, characterized by their velocity and their shift. Given the parameters of K K solitons R k R^k (with distinct velocities), we prove the existence and uniqueness of a multi-soliton u u such that \[ ‖ u ( t ) − ∑ k = 1 K R k ( t ) ‖ H 1 → 0 as t → + ∞ . \| u(t) - \sum _{k=1}^K R^k(t) \|_{H^1} \to 0 \quad \text {as} \quad t \to +\infty . \] The convergence takes place in H s H^s with an exponential rate for all s ≥ 0 s \ge 0 . The construction is made by successive approximations of the multi-soliton. We use classical arguments to control of H 1 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 H^s -norm in dimension d ≥ 2 d\geq 2 , by a technique close to monotonicity.