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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Asymptotic $K$-soliton-like solutions of the Zakharov-Kuznetsov type equations
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by Frédéric Valet PDF
Trans. Amer. Math. Soc. 374 (2021), 3177-3213 Request permission

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.
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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
  • Received by editor(s): June 2, 2020
  • Published electronically: March 8, 2021
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 3177-3213
  • MSC (2020): Primary 35Q53, 35Q35, 35B40, 37K40
  • DOI: https://doi.org/10.1090/tran/8331
  • MathSciNet review: 4237946