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July  2021, 26(7): 3479-3490. doi: 10.3934/dcdsb.2020241

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

Received  April 2019 Published  July 2021 Early access  August 2020

Fund Project: 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)

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.
Keywords: Wave equation, Liouvillean frequency, quasi-periodic solution, infinite dimensional KAM theory, Hamiltonian system, fractional Schrödinger equation system, Töplitz-Lipschitz property, normal form.
Mathematics Subject Classification: Primary:35Q35;37K55.
Citation: Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241
References:
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A. Avila, Almost reducitility and absolute continuity, preprint, arXiv: 1006.0704. Google Scholar

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J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439.  doi: 10.2307/121001.  Google Scholar

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J. Bourgain, Nonlinear Schrödinger Equations, Park City Ser., 5, American Mathematical Society, Providence, 1999. doi: 10.1090/coll/046.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119â€"148.  Google Scholar

[27]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.  doi: 10.1007/BF02566420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

X. Xu, J. You and Q. Zhou, Quasi-periodic solutions of NLS with Liouvillean frequency, preprint, arXiv: 1707.04048. Google Scholar

show all references

References:
[1]

A. Avila, Almost reducitility and absolute continuity, preprint, arXiv: 1006.0704. Google Scholar

[2]

A. Avila, Global theory of one-frequency Schrödinger operators, Acta Math., 215 (2015), 1-54.  doi: 10.1007/s11511-015-0128-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Bourgain, Nonlinear Schrödinger Equations, Park City Ser., 5, American Mathematical Society, Providence, 1999. doi: 10.1090/coll/046.  Google Scholar

[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.  Google Scholar
[12]

J. Bourgain, On Melnikov's persistency problem, Math. Res. Lett., 4 (1997), 445-458.  doi: 10.4310/MRL.1997.v4.n4.a1.  Google Scholar

[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.  Google Scholar

[14]

L. Eliasson, Perturbations of stable invariant tori, Ann. Sc. Norm. Sup. Pisa CI Sci. Iv Ser., 15 (1998), 115-147.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

T. Kappeler and J. Pöschel, KDV & KAM, Spinger, Berlin, 1993. doi: 10.1007/978-3-662-08054-2.  Google Scholar

[20]

S. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0092243.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119â€"148.  Google Scholar

[27]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.  doi: 10.1007/BF02566420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

X. Xu, J. You and Q. Zhou, Quasi-periodic solutions of NLS with Liouvillean frequency, preprint, arXiv: 1707.04048. Google Scholar

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