Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Frédéric Valet
Journal: Trans. Amer. Math. Soc. 374 (2021), 3177-3213
MSC (2020): Primary 35Q53, 35Q35, 35B40, 37K40
DOI: https://doi.org/10.1090/tran/8331
Published electronically: March 8, 2021
MathSciNet review: 4237946
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 35Q53, 35Q35, 35B40, 37K40

Retrieve articles in all journals with MSC (2020): 35Q53, 35Q35, 35B40, 37K40


Additional Information

Frédéric Valet
Affiliation: IRMA UMR 7501, Université de Strasbourg, CNRS, F-67000 Strasbourg, France
Address at time of publication: Postboks 7803, 5020 Bergen, Norway
Email: frederic.valet@uib.no

Keywords: Zakharov-Kuznetsov equation, multi-soliton
Received by editor(s): June 2, 2020
Published electronically: March 8, 2021
Article copyright: © Copyright 2021 American Mathematical Society