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.