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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 and Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241
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.

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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.

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J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 629-639.  doi: 10.1007/BF01902055.

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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.

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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.
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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.

show all references

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.

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