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Singular perturbations and scaling
Quasi-periodic solutions for a class of beam equation system
1. | College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China |
2. | Department of Mathematics, Southeast University, Nanjing 211189, China |
$ \left\{ \begin{array}{lll} u_{1tt}+ \Delta^2 u_1 +\sigma u_1 +u_1u_2^2 & = & 0 \\ &&\\ u_{2tt}+ \Delta^2 u_2 +\mu u_2 +u_1^2 u_2 & = & 0 \end{array} \right. $ |
$ 0<\sigma \in [ \sigma_1,\sigma_2 ], $ |
$ 0<\mu\in [ \mu_1,\mu_2 ] $ |
References:
[1] |
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. |
[2] |
D. Bambusi,
On long time stability in Hamiltonian perturbations of non-resonant linear PDEs, Nonlinearity, 12 (1999), 823-850.
doi: 10.1088/0951-7715/12/4/305. |
[3] |
M. Berti and P. Bolle,
Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity, 25 (2012), 2579-2613.
doi: 10.1088/0951-7715/25/9/2579. |
[4] |
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. |
[5] |
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. |
[6] |
J. Bourgain,
Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 629-639.
doi: 10.1007/BF01902055. |
[7] |
J. Bourgain,
Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[8] |
J. Bourgain, Nonlinear Schrödinger equations, Hyperbolic Equations and Frequency Interactions (Park City, UT, 1995), 3–157, IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, 1999.
doi: 10.1090/coll/046. |
[9] |
J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Princeton Univ. Press, Princeton, 2005.
doi: 10.1515/9781400837144.![]() ![]() ![]() |
[10] |
W. Craig and C. E. Wayne,
Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498.
doi: 10.1002/cpa.3160461102. |
[11] |
L. H. Eliasson and S. B. Kuksin,
KAM for the nonlinear Schrödinger equation, Ann. of Math., 172 (2010), 371-435.
doi: 10.4007/annals.2010.172.371. |
[12] |
J. Geng, X. Xu and J. You,
An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., 226 (2011), 5361-5402.
doi: 10.1016/j.aim.2011.01.013. |
[13] |
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. |
[14] |
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. |
[15] |
J. Geng and J. You,
A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys, 262 (2006), 343-372.
doi: 10.1007/s00220-005-1497-0. |
[16] |
J. Geng and J. You,
KAM tori for higher dimensional beam equations with constant potentials, Nonlinearity, 19 (2006), 2405-2423.
doi: 10.1088/0951-7715/19/10/007. |
[17] |
B. Grebert and V. Rocha, Stable and unstable time quasi periodic solutions for a system of coupled NLS equations, 2018 IOP Publishing Ltd & London Mathematical Society, 31 (2018), arXiv: 1710.09173v1.
doi: 10.1088/1361-6544/aad3d9. |
[18] |
S. B. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993.
doi: 10.1007/BFb0092243. |
[19] |
S. B. 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. |
[20] |
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. |
[21] |
J. Pöschel,
A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.
|
[22] |
J. Pöschel,
Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.
doi: 10.1007/BF02566420. |
[23] |
M. Procesi and X. Xu,
Quasi-Töplitz functions in KAM theorem, SIAM J. Math. Anal., 45 (2013), 2148-2181.
doi: 10.1137/110833014. |
[24] |
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. |
[25] |
Y. Shi, J. Xu and X. Xu,
Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing, Discrete and Continuous Dynamical System-B, 22 (2017), 2501-2519.
doi: 10.3934/dcdsb.2017104. |
[26] |
Y. Shi, X. Lu and X. Xu,
Quasi-periodic solutions for Schrödinger equation with derivative nonlinearity, Dynamical Systems, 30 (2015), 158-188.
doi: 10.1080/14689367.2014.993924. |
[27] |
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. |
[28] |
J. Xu and J. You,
Persistence of lower-dimensional tori under the first Melnikov's non-resnonce condition, J. Math. Pures Appl., 80 (2001), 1045-1067.
doi: 10.1016/S0021-7824(01)01221-1. |
[29] |
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. |
[30] |
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. |
[31] |
S. Zhou, An abstract infinite dimensional KAM theorem with application to nonlinear higher dimensional Schrödinger equation systems, arXiv: 1701.05727v1. |
show all references
References:
[1] |
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. |
[2] |
D. Bambusi,
On long time stability in Hamiltonian perturbations of non-resonant linear PDEs, Nonlinearity, 12 (1999), 823-850.
doi: 10.1088/0951-7715/12/4/305. |
[3] |
M. Berti and P. Bolle,
Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity, 25 (2012), 2579-2613.
doi: 10.1088/0951-7715/25/9/2579. |
[4] |
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. |
[5] |
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. |
[6] |
J. Bourgain,
Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 629-639.
doi: 10.1007/BF01902055. |
[7] |
J. Bourgain,
Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[8] |
J. Bourgain, Nonlinear Schrödinger equations, Hyperbolic Equations and Frequency Interactions (Park City, UT, 1995), 3–157, IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, 1999.
doi: 10.1090/coll/046. |
[9] |
J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Princeton Univ. Press, Princeton, 2005.
doi: 10.1515/9781400837144.![]() ![]() ![]() |
[10] |
W. Craig and C. E. Wayne,
Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498.
doi: 10.1002/cpa.3160461102. |
[11] |
L. H. Eliasson and S. B. Kuksin,
KAM for the nonlinear Schrödinger equation, Ann. of Math., 172 (2010), 371-435.
doi: 10.4007/annals.2010.172.371. |
[12] |
J. Geng, X. Xu and J. You,
An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., 226 (2011), 5361-5402.
doi: 10.1016/j.aim.2011.01.013. |
[13] |
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. |
[14] |
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. |
[15] |
J. Geng and J. You,
A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys, 262 (2006), 343-372.
doi: 10.1007/s00220-005-1497-0. |
[16] |
J. Geng and J. You,
KAM tori for higher dimensional beam equations with constant potentials, Nonlinearity, 19 (2006), 2405-2423.
doi: 10.1088/0951-7715/19/10/007. |
[17] |
B. Grebert and V. Rocha, Stable and unstable time quasi periodic solutions for a system of coupled NLS equations, 2018 IOP Publishing Ltd & London Mathematical Society, 31 (2018), arXiv: 1710.09173v1.
doi: 10.1088/1361-6544/aad3d9. |
[18] |
S. B. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993.
doi: 10.1007/BFb0092243. |
[19] |
S. B. 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. |
[20] |
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. |
[21] |
J. Pöschel,
A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.
|
[22] |
J. Pöschel,
Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.
doi: 10.1007/BF02566420. |
[23] |
M. Procesi and X. Xu,
Quasi-Töplitz functions in KAM theorem, SIAM J. Math. Anal., 45 (2013), 2148-2181.
doi: 10.1137/110833014. |
[24] |
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. |
[25] |
Y. Shi, J. Xu and X. Xu,
Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing, Discrete and Continuous Dynamical System-B, 22 (2017), 2501-2519.
doi: 10.3934/dcdsb.2017104. |
[26] |
Y. Shi, X. Lu and X. Xu,
Quasi-periodic solutions for Schrödinger equation with derivative nonlinearity, Dynamical Systems, 30 (2015), 158-188.
doi: 10.1080/14689367.2014.993924. |
[27] |
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. |
[28] |
J. Xu and J. You,
Persistence of lower-dimensional tori under the first Melnikov's non-resnonce condition, J. Math. Pures Appl., 80 (2001), 1045-1067.
doi: 10.1016/S0021-7824(01)01221-1. |
[29] |
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. |
[30] |
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. |
[31] |
S. Zhou, An abstract infinite dimensional KAM theorem with application to nonlinear higher dimensional Schrödinger equation systems, arXiv: 1701.05727v1. |
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