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Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency
1. | College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China |
2. | Department of Mathematics, Southeast University, Nanjing 211189, China |
$ u_{tt}-u_{xx} +mu +\varepsilon f(\omega t,x,u;\xi) = 0 $ |
$ \varepsilon $ |
$ \omega = \xi \bar{\omega}, $ |
$ \bar{\omega} $ |
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. |
[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. 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. |
[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. Google Scholar |
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