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January  2020, 25(1): 31-53. doi: 10.3934/dcdsb.2019171

Quasi-periodic solutions for a class of beam equation system

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: shiyanling96998@163.com

Received  October 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author is partially supported by NSFC Grant(11801492, 61877052), NSFJS Grant (BK 20170472) and NSF of Jiangsu Higher education Institute of China Grant(18KJB110030). The second author is supported by the NSFC Grant(11871146).

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional beam equation system
$ \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. $
under periodic boundary conditions, where
$ 0<\sigma \in [ \sigma_1,\sigma_2 ], $
$ 0<\mu\in [ \mu_1,\mu_2 ] $
 are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.
Keywords: Beam equation system, quasi-periodic solution, infinite dimensional KAM theory.
Mathematics Subject Classification: Primary: 37K55; Secondary: 35G30.
Citation: Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9] J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Princeton Univ. Press, Princeton, 2005.  doi: 10.1515/9781400837144.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[23]

M. Procesi and X. Xu, Quasi-Töplitz functions in KAM theorem, SIAM J. Math. Anal., 45 (2013), 2148-2181.  doi: 10.1137/110833014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. Zhou, An abstract infinite dimensional KAM theorem with application to nonlinear higher dimensional Schrödinger equation systems, arXiv: 1701.05727v1. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9] J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Princeton Univ. Press, Princeton, 2005.  doi: 10.1515/9781400837144.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[23]

M. Procesi and X. Xu, Quasi-Töplitz functions in KAM theorem, SIAM J. Math. Anal., 45 (2013), 2148-2181.  doi: 10.1137/110833014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. Zhou, An abstract infinite dimensional KAM theorem with application to nonlinear higher dimensional Schrödinger equation systems, arXiv: 1701.05727v1. Google Scholar

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