Quasi-periodic solutions of Sawada-Kotera equation

Wenlei Li; Juanying Huo; Wenlei Li; Juanying Huo

doi:10.3934/math.2026237

AIMS Mathematics

2026, Volume 11, Issue 3: 5759-5775. doi: 10.3934/math.2026237

Previous Article Next Article

Research article Special Issues

Quasi-periodic solutions of Sawada-Kotera equation

Wenlei Li ,
Juanying Huo ^,

School of Mathematics, Jilin University, Changchun, China

Received: 12 December 2025 Revised: 04 February 2026 Accepted: 24 February 2026 Published: 09 March 2026
MSC : 35L76, 70K43, 70K75

This paper establishes the existence of quasi-periodic solutions to the Sawada-Kotera equation on the torus $\mathbb{T}$. By implementing a Nash-Moser iterative method, we demonstrate that for a prescribed Diophantine frequency vector $\lambda \tilde{\omega}$, such solutions exist for a set of parameters $\lambda$ of positive Lebesgue measure.
- Sawada-Kotera equation,
- quasi-periodic solutions,
- Nash-Moser iteration,
- small divisor problem,
- Hamiltonian PDEs
Citation: Wenlei Li, Juanying Huo. Quasi-periodic solutions of Sawada-Kotera equation[J]. AIMS Mathematics, 2026, 11(3): 5759-5775. doi: 10.3934/math.2026237

Related Papers:

Abstract

This paper establishes the existence of quasi-periodic solutions to the Sawada-Kotera equation on the torus $\mathbb{T}$. By implementing a Nash-Moser iterative method, we demonstrate that for a prescribed Diophantine frequency vector $\lambda \tilde{\omega}$, such solutions exist for a set of parameters $\lambda$ of positive Lebesgue measure.

References

[1]	M. Berti, P. Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on $\Bbb T^d$ with a multiplicative potential, J. Eur. Math. Soc. (JEMS), 15 (2013), 229–286. https://doi.org/10.4171/JEMS/361 doi: 10.4171/JEMS/361
[2]	M. Berti, Z. Hassainia, N. Masmoudi, Time quasi-periodic vortex patches of Euler equation in the plane, Invent. Math., 233 (2023), 1279–1391. https://doi.org/10.1007/s00222-023-01195-4 doi: 10.1007/s00222-023-01195-4
[3]	J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. Math., 148 (1998), 363–439. https://doi.org/10.2307/121001 doi: 10.2307/121001
[4]	W. Craig, C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409–1498. https://doi.org/10.1002/cpa.3160461102 doi: 10.1002/cpa.3160461102
[5]	L. H. Eliasson, S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math., 172 (2010), 371–435. https://doi.org/10.4007/annals.2010.172.371 doi: 10.4007/annals.2010.172.371
[6]	L. Franzoi, R. Montalto, A KAM approach to the inviscid limit for the 2D Navier-Stokes equations, Ann. Henri Poincaré, 25 (2024), 5231–5275. https://doi.org/10.1007/s00023-023-01408-9 doi: 10.1007/s00023-023-01408-9
[7]	K. A. Gorshkov, L. A. Ostrovsky, V. V. Papko, A. S. Pikovsky, On the existence of stationary multisolitons, Phys. Lett. A, 74 (1979), 177–179. https://doi.org/10.1016/0375-9601(79)90763-1 doi: 10.1016/0375-9601(79)90763-1
[8]	A. K. Gupta, S. Saha Ray, Numerical treatment for the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method, Appl. Math. Model., 39 (2015), 5121–5130. https://doi.org/10.1016/j.apm.2015.04.003 doi: 10.1016/j.apm.2015.04.003
[9]	Z. Hassainia, T. Hmidi, E. Roulley, Invariant KAM tori around annular vortex patches for 2D Euler equations, Commun. Math. Phys., 405 (2024), 270. https://doi.org/10.1007/s00220-024-05141-0 doi: 10.1007/s00220-024-05141-0
[10]	T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Jpn., 33 (1972), 260–264. https://doi.org/10.1143/jpsj.33.260 doi: 10.1143/jpsj.33.260
[11]	S. Kuksin, J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. Math., 143 (1996), 149–179. https://doi.org/10.2307/2118656 doi: 10.2307/2118656
[12]	K. Sawada, T. Kotera, A method for finding $N$-soliton solutions of the K.d.V. equation and K.d.V.-like equation, Prog. Theor. Phys., 51 (1974), 1355–1367. https://doi.org/10.1143/ptp.51.1355 doi: 10.1143/ptp.51.1355
[13]	M. E. Taylor, Partial differential equations III. Nonlinear equations, Applied Mathematical Sciences, Vol. 117, Springer, Cham, 2023. https://doi.org/10.1007/978-3-031-33928-8
[14]	A. Yusuf, T. A. Sulaiman, Dynamics of lump-periodic, breather and two-wave solutions with the long wave in shallow water under gravity and 2D nonlinear lattice, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), 105846. https://doi.org/10.1016/j.cnsns.2021.105846 doi: 10.1016/j.cnsns.2021.105846

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)