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

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) − ∑_k=1^K R^k(t)|_H1 → 0 as t → + ∞. The convergence takes place in H^s with an exponential rate for all s ≥ 0. The construction is made by successive approximations of the multi-soliton. We use classical arguments to control of H¹-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 ≥ 2, by a technique close to monotonicity. [ABSTRACT FROM AUTHOR]