Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency

Yanling Shi; Junxiang Xu

doi:10.3934/dcdsb.2020241

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2021, Volume 26, Issue 7: 3479-3490. Doi: 10.3934/dcdsb.2020241

This issue Previous Article Next Article Moran process and Wright-Fisher process favor low variability

Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency

Yanling Shi^1, , and
Junxiang Xu^2,

1.
College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China
2.
Department of Mathematics, Southeast University, Nanjing 211189, China

^* Corresponding author: Yanling Shi
^* Corresponding author: Yanling Shi

Received: March 31, 2019

Early access: August 2020

Published: July 2021

The first author is partially supported by NSFC Grant(11801492, 61877052), NSFJS Grant (BK 20170472). The second author is supported by the NSFC Grant(11871146)

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Abstract

In this paper, one dimensional nonlinear wave equation

$ u_{tt}-u_{xx} +mu +\varepsilon f(\omega t,x,u;\xi) = 0 $

with Dirichlet boundary condition is considered, where $ \varepsilon $ is small positive parameter, $ \omega = \xi \bar{\omega}, $ $ \bar{\omega} $ is weak Liouvillean frequency. It is proved that there are many quasi-periodic solutions with Liouvillean frequency for the above equation. The proof is based on an infinite dimensional KAM Theorem.

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Mathematics Subject Classification: Primary:35Q35;37K55.

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References

[1]	A. Avila, Almost reducitility and absolute continuity, preprint, arXiv: 1006.0704.
[2]	A. Avila, Global theory of one-frequency Schrödinger operators, Acta Math., 215 (2015), 1-54. doi: 10.1007/s11511-015-0128-7.
[3]	A. Avila, B. Fayad and R. Krikorian, A KAM scheme for $\mathbb{SL}(2,\mathbb{R})$ cocycles with Liouvillean frequencies, Geom. Funct. Anal., 21 (2011), 1001-1019. doi: 10.1007/s00039-011-0135-6.
[4]	M. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480. doi: 10.1007/s002200100426.
[5]	M. Berti and L. Biasco, Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys., 305 (2011), 741-796. doi: 10.1007/s00220-011-1264-3.
[6]	M. Berti and P. Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb{T}^d$ with a multiplicative potential, Eur. J. Math., 15 (2013), 229-286. doi: 10.4171/JEMS/361.
[7]	J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices, 11 (1994), 475-497. doi: 10.1155/S1073792894000516.
[8]	J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 629-639. doi: 10.1007/BF01902055.
[9]	J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439. doi: 10.2307/121001.
[10]	J. Bourgain, Nonlinear Schrödinger Equations, Park City Ser., 5, American Mathematical Society, Providence, 1999. doi: 10.1090/coll/046.
[11]	J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158, Princeton Univ. Press, 2005. doi: 10.1515/9781400837144.
[12]	J. Bourgain, On Melnikov's persistency problem, Math. Res. Lett., 4 (1997), 445-458. doi: 10.4310/MRL.1997.v4.n4.a1.
[13]	W. Craig and C. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102.
[14]	L. Eliasson, Perturbations of stable invariant tori, Ann. Sc. Norm. Sup. Pisa CI Sci. Iv Ser., 15 (1998), 115-147.
[15]	J. Geng and Y. Yi, Quasi-periodic solutions in a nonlinear Schrödinger equation, J. Differential Equations, 233 (2007), 512-542. doi: 10.1016/j.jde.2006.07.027.
[16]	J. Geng and J. You, A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions, J. Differential Equations, 209 (2005), 1-56. doi: 10.1016/j.jde.2004.09.013.
[17]	J. Geng and X. Ren, Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation, J. Diff. Eq., 249 (2010), 2796-2821. doi: 10.1016/j.jde.2010.04.003.
[18]	X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi periodic linear sysems, Invent. Math., 190 (2012), 209-260. doi: 10.1007/s00222-012-0379-2.
[19]	T. Kappeler and J. Pöschel, KDV & KAM, Spinger, Berlin, 1993. doi: 10.1007/978-3-662-08054-2.
[20]	S. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0092243.
[21]	S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149-179. doi: 10.2307/2118656.
[22]	R. Krikorian, J. Wang, J. You and Q. Zhou, Linearization of quasi periodically forced circle flow beyond brjuno condition, Comm. Math. Phys., 358 (2018), 81-100. doi: 10.1007/s00220-017-3021-8.
[23]	Z. Liang and J. You, Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity, SIAM J. Math. Anal., 36 (2005), 1965-1990. doi: 10.1137/S0036141003435011.
[24]	J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equation with unbounded perturbations, Comm. Math. Phys., 307 (2011), 629-673. doi: 10.1007/s00220-011-1353-3.
[25]	H. Niu and J. Geng, Almost periodic solutions for a class of higher dimensional beam equations, Nonlinearity, 20 (2007), 2499-2517. doi: 10.1088/0951-7715/20/11/003.
[26]	J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119â€"148.
[27]	J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.
[28]	Y. Shi, J. Xu and X. Xu, On quasi-periodic solutions for a generalized Boussinesq equation, Nonlinear Anal., 105 (2014), 50-61. doi: 10.1016/j.na.2014.04.007.
[29]	C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528. doi: 10.1007/BF02104499.
[30]	X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations., 230 (2006), 213-274. doi: 10.1016/j.jde.2005.12.012.
[31]	M. Zhang and J. Si, Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing, Phys. D, 238 (2009), 2185-2215. doi: 10.1016/j.physd.2009.09.003.
[32]	X. Xu, J. You and Q. Zhou, Quasi-periodic solutions of NLS with Liouvillean frequency, preprint, arXiv: 1707.04048.