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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

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.
Citation: Guanghua Shi, Dongfeng Yan. KAM tori for quintic nonlinear schrödinger equations with given potential. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2421-2439. doi: 10.3934/dcds.2020120
References:
[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. Kuksin, Analysis 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. Trubowitz, Inverse 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. Titchmarsh, Eigenfunction 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.

show all references

References:
[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. Kuksin, Analysis 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. Trubowitz, Inverse 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. Titchmarsh, Eigenfunction 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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