# American Institute of Mathematical Sciences

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

## 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

* Corresponding author: shiyanling96998@163.com

Received  October 2018 Revised  March 2019 Published  January 2020 Early access  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.
Citation: Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete and 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. [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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##### 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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