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April  2020, 40(4): 2421-2439. doi: 10.3934/dcds.2020120

KAM tori for quintic nonlinear schrödinger equations with given potential

Guanghua Shi 1, and  Dongfeng Yan 2,,

1. 

College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China
2. 

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, 450001, China

* Corresponding author: Dongfeng Yan

Received  July 2019 Revised  November 2019 Published  January 2020

Fund Project: The first author is supported by NSFC grant 11371132, and the second author is supported by NSFC grant 11601487.

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.
Keywords: KAM theory, quasi-periodic solutions, Birkhoff normal form.
Mathematics Subject Classification: Primary: 37K55; Secondary: 37J40.
Citation: Guanghua Shi, Dongfeng Yan. KAM tori for quintic nonlinear schrödinger equations with given potential. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2421-2439. doi: 10.3934/dcds.2020120
References:
[1]

P. Baldi, M. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439.  doi: 10.2307/121001.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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T. Kappler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, Heidelberg, 2003. doi: 10.1007/978-3-662-08054-2.  Google Scholar

[12]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen., 21 (1987), 22–37, 95.  Google Scholar

[13]

S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR Izv., 32 (1989), 39-62.  doi: 10.1070/IM1989v032n01ABEH000733.  Google Scholar

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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.  Google Scholar

[16]

S. B. Kuksin, A KAM theorem for equations of the Korteweg-de Vries type, Rev. Math-Math Phys., 10 (1998), 1-64.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[23] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000.   Google Scholar
[24]

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

[25]

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

[26] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part Ⅰ, Second Edition, Clarendon Press, Oxford, 1962.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. Zhang, M. 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.  Google Scholar

show all references

References:
[1]

P. Baldi, M. 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.  Google Scholar

[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.  Google Scholar

[3]

M. Berti, L. 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.  Google Scholar

[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.  Google Scholar

[5]

J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439.  doi: 10.2307/121001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. Kappler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, Heidelberg, 2003. doi: 10.1007/978-3-662-08054-2.  Google Scholar

[12]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen., 21 (1987), 22–37, 95.  Google Scholar

[13]

S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR Izv., 32 (1989), 39-62.  doi: 10.1070/IM1989v032n01ABEH000733.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

S. B. Kuksin, A KAM theorem for equations of the Korteweg-de Vries type, Rev. Math-Math Phys., 10 (1998), 1-64.   Google Scholar

[17] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000.   Google Scholar
[24]

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

[25]

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

[26] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part Ⅰ, Second Edition, Clarendon Press, Oxford, 1962.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. Zhang, M. 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.  Google Scholar

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