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KAM tori for quintic nonlinear schrödinger equations with given potential

  • * Corresponding author: Dongfeng Yan

    * Corresponding author: Dongfeng Yan

The first author is supported by NSFC grant 11371132, and the second author is supported by NSFC grant 11601487

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  • This paper is concerned with the 1-dimensional quintic nonlinear Schrödinger equations with real valued $ C^{\infty} $-smooth given potential

    $ \sqrt{-1}u_{t} = u_{xx}-V(x)u-|u|^4u $

    subject to Dirichlet boundary conditions. By means of normal form theory and an infinite-dimensional Kolmogorov-Arnold-Moser (KAM, for short) theorem, it is proved that the above equation admits a family of elliptic tori where lies small amplitude quasi-periodic solutions with two frequencies of high modes.

    Mathematics Subject Classification: Primary: 37K55; Secondary: 37J40.

    Citation:

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  • [1] P. BaldiM. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536.  doi: 10.1007/s00208-013-1001-7.
    [2] D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., 135 (2006), 507-567.  doi: 10.1215/S0012-7094-06-13534-2.
    [3] M. BertiL. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equations, Arch. Ration. Mech. Anal., 212 (2014), 905-955.  doi: 10.1007/s00205-014-0726-0.
    [4] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int. Math. Res. Not., 1994 (1994), 475-497.  doi: 10.1155/S1073792894000516.
    [5] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439.  doi: 10.2307/121001.
    [6] J. Bourgain, On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal., 229 (2005), 62-94.  doi: 10.1016/j.jfa.2004.10.019.
    [7] C. M. Cao and X. P. Yuan, Quasi-peiodic solutions for perpertued generalized nonlinear vibrating string equation with singularities, Discrete Contin. Dyn. Syst., 37 (2017), 1867-1901.  doi: 10.3934/dcds.2017079.
    [8] L. Chierchia and J. G. You, KAM tori for 1D nonlinear wave equation with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525.  doi: 10.1007/s002200050824.
    [9] L. J. Du and X. P. Yuan, Invariant tori for nonlinear Schrödinger equations with a given potential, Dynamics of PDE, 3 (2006), 331-346.  doi: 10.4310/DPDE.2006.v3.n4.a4.
    [10] M. N. Gao and J. J. Liu, Quasi-periodic solutions for 1D wave equation with higher order nonlinearity, J. Differential Equations, 252 (2012), 1466-1493.  doi: 10.1016/j.jde.2011.10.006.
    [11] T. Kappler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, Heidelberg, 2003. doi: 10.1007/978-3-662-08054-2.
    [12] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen., 21 (1987), 22–37, 95.
    [13] S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR Izv., 32 (1989), 39-62.  doi: 10.1070/IM1989v032n01ABEH000733.
    [14] S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556. Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0092243.
    [15] S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. Math., 143 (1996), 149-179.  doi: 10.2307/2118656.
    [16] S. B. Kuksin, A KAM theorem for equations of the Korteweg-de Vries type, Rev. Math-Math Phys., 10 (1998), 1-64. 
    [17] S. B. KuksinAnalysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000. 
    [18] Z. G. Liang and J. G. You, Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity, SIAM J. Math. Anal., 36 (2005), 1965-1990.  doi: 10.1137/S0036141003435011.
    [19] J. J. Liu and X. P. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient, Commun. Pure Appl. Math., 63 (2010), 1145-1172.  doi: 10.1002/cpa.20314.
    [20] J. J. Liu and X. P. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Commun. Math. Phys., 307 (2011), 629-673.  doi: 10.1007/s00220-011-1353-3.
    [21] J. J. Liu and X. P. Yuan, KAM for the derivative nonliear Schrödinger equation with periodic boundary conditions, J. Differential Equations, 256 (2014), 1627-1652.  doi: 10.1016/j.jde.2013.11.007.
    [22] L. F. Mi, Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential, J. Math. Anal. Appl., 390 (2012), 335-354.  doi: 10.1016/j.jmaa.2012.01.046.
    [23] J. Pöschel and  E. TrubowitzInverse Spectral Theory, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000. 
    [24] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148. 
    [25] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.  doi: 10.1007/BF02566420.
    [26] E. C. TitchmarshEigenfunction Expansions Associated with Second-Order Differential Equations. Part Ⅰ, Second Edition, Clarendon Press, Oxford, 1962. 
    [27] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127 (1990), 479-528.  doi: 10.1007/BF02104499.
    [28] X. P. 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.
    [29] X. P. Yuan, Invatiant tori of nonlinear wave equations with a given potential, Discrete Contin. Dyn. Syst., 16 (2006), 615-634.  doi: 10.3934/dcds.2006.16.615.
    [30] J. ZhangM. N. Gao and X. P. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1189-1228.  doi: 10.1088/0951-7715/24/4/010.
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