KAM tori for quintic nonlinear schrödinger equations with given potential

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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 and Continuous Dynamical Systems, 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.
[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. 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.
[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. 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.

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.
[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. 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.
[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. 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.

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