January  2017, 16(1): 25-68. doi: 10.3934/cpaa.2017002

Invariant tori of a nonlinear Schrödinger equation with quasi-periodically unbounded perturbations

1. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, P. R. China

2. 

School of Mathematics, Shandong University, Jinan 250100, P. R. China

Jianguo Si, E-mail address: sijgmath@sdu.edu.cn

Received  June 2015 Revised  August 2016 Published  November 2016

Fund Project: The first author is supported by the National Natural Science Foundation of China (Grant 11171185, 11571201) and the Research Foundation for Doctor Programme of Henan Polytechnic University (Grant B2016-58). the second author is supported by the National Natural Science Foundation of China (Grant 11171185, 11571201).

This paper is concerned with the derivative nonlinear Schrödinger equation with quasi-periodic forcing under periodic boundary conditions
$ {\rm{i}} u_t+u_{xx}+{\rm{i}} (B+\epsilon g(\beta t))(f(|u|^2)u)_x=0, \ \ x\in \mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}. $
Assume that the frequency vector $\beta$ is co-linear with a fixed Diophantine vector $\bar{\beta}\in \mathbb{R}.{m}$, that is, $\beta=\lambda \bar{\beta}$, $\lambda \in [1/2, 3/2]$. We show that above equation possesses a Cantorian branch of invariant $n$--tori and exists many smooth quasi-periodic solutions with $(m+n)$ non-resonance frequencies $(\lambda\bar{\beta}, \omega_{\ast})$. The proof is based on a Kolmogorov--Arnold--Moser (KAM) iterative procedure for quasi-periodically unbounded vector fields and partial Birkhoff normal form.
Citation: Jie Liu, Jianguo Si. Invariant tori of a nonlinear Schrödinger equation with quasi-periodically unbounded perturbations. Communications on Pure and Applied Analysis, 2017, 16 (1) : 25-68. doi: 10.3934/cpaa.2017002
References:
[1]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbation of Airy equation, Math. Ann., 359 (2014), 471-536. doi: 10.1007/s00208-013-1001-7.

[2]

M. Berti and L. Biasco, Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys., 305 (2011), 741-796. doi: 10.1007/s00220-011-1264-3.

[3]

M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Eqautions, 31 (2006), 959-985. doi: 10.1080/03605300500358129.

[4]

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

[5]

J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158, Princeton University Press, Princeton, 2005. doi: 10.1515/9781400837144.

[6]

L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824.

[7]

W. Craig and C. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102.

[8]

H. L. Eliasson and S. B. Kuksin, KAM for the non-linear Schrödinger equation, Ann. of Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371.

[9]

R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations, 259 (2015), 3389-3447. doi: 10.1016/j.jde.2015.04.025.

[10]

R. Feola, KAM for quasi-linear forced Hamiltonian NLS, preprint, arXiv: 1602.01341.

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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.

[12]

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.

[13]

J. Geng, Invariant tori of full dimension for a nonlinear Schrödinger equation, J. Differential Equations, 252 (2012), 1-34. doi: 10.1016/j.jde.2011.09.006.

[14]

J. Geng and J. Wu, Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations, J. Math. Phys., 53 (2012), 102702. doi: 10.1063/1.4754822.

[15]

B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys., 307 (2011), 383-427. doi: 10.1007/s00220-011-1327-5.

[16]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, (Russian) Funktsional. Anal. iPrilozhen., 21 (1987), 22-37; English translation in: Funct. Anal. Appl., 21 (1987), 192-205.

[17]

S. B. Kuksin, Perturbations of conditionally periodic solutions of infinite-dimensional Hamiltonian systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41-63; English translation in: Math. USSR-Izv., 32 (1989), 39-62.

[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), 147-179.

[20]

S. B. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271. doi: 10.1007/PL00001476.

[21]

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

[22]

T. Kappeler and J. Pöschel, KdV and KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.

[23]

L. Jiao and Y. Wang, The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation, Commun. Pure Appl. Anal., 8 (2009), 1585-1606. doi: 10.3934/cpaa.2009.8.1585.

[24]

J. Liu and J. Si, Invariant tori for a derivative nonlinear Schrödinger equation with quasiperiodic forcing, J Math. Phys. , 56 (2015), 032702. doi: 10.1063/1.4916287.

[25]

J. Liu, Periodic and quasi-periodic solutions of a derivative nonlinear Schrödinger equation, Appl. Anal., 95 (2016), 801-825. doi: 10.1080/00036811.2015.1032942.

[26]

J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient, Comm. Pure. Appl. Math., 63 (2010), 1145-1172. doi: 10.1002/cpa.20314.

[27]

J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys., 307 (2011), 629-673. doi: 10.1007/s00220-011-1353-3.

[28]

J. Liu and X. Yuan, KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions, J. Differential Equations, 256 (2014), 1627-1652. doi: 10.1016/j.jde.2013.11.007.

[29]

L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasiperiodically forced perturbation, Discrete Contin. Dyn. Syst., 34 (2014), 689-707. doi: 10.3934/dcds.2014.34.689.

[30]

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

[31]

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

[32]

J. Rui and J. Si, Quasi-periodic solutions for quasi-periodically forced nonlinear Schrödinger equations with quasi-periodic inhomogeneous terms, Phys. D, 286/287 (2014), 1-31. doi: 10.1016/j.physd.2014.07.005.

[33]

M. B. Sevryuk, The reversible context 2 in KAM theory: the first steps, Regul. Chaotic Dyn., 16 (2011), 24-38. doi: 10.1134/S1560354710520035.

[34]

J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing, J. Differential Equations, 252 (2012), 5274-5360. doi: 10.1016/j.jde.2012.01.034.

[35]

Y. Wang and J. Si, A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term, Z. Angew. Math. Phys., 63 (2012), 189-190. doi: 10.1007/s00033-011-0172-x.

[36]

C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528.

[37]

X. Yuan, Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension, J. Differential Equations, 195 (2003), 230-242. doi: 10.1016/S0022-0396(03)00095-0.

[38]

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.

[39]

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.

show all references

References:
[1]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbation of Airy equation, Math. Ann., 359 (2014), 471-536. doi: 10.1007/s00208-013-1001-7.

[2]

M. Berti and L. Biasco, Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys., 305 (2011), 741-796. doi: 10.1007/s00220-011-1264-3.

[3]

M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Eqautions, 31 (2006), 959-985. doi: 10.1080/03605300500358129.

[4]

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

[5]

J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158, Princeton University Press, Princeton, 2005. doi: 10.1515/9781400837144.

[6]

L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824.

[7]

W. Craig and C. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102.

[8]

H. L. Eliasson and S. B. Kuksin, KAM for the non-linear Schrödinger equation, Ann. of Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371.

[9]

R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations, 259 (2015), 3389-3447. doi: 10.1016/j.jde.2015.04.025.

[10]

R. Feola, KAM for quasi-linear forced Hamiltonian NLS, preprint, arXiv: 1602.01341.

[11]

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.

[12]

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.

[13]

J. Geng, Invariant tori of full dimension for a nonlinear Schrödinger equation, J. Differential Equations, 252 (2012), 1-34. doi: 10.1016/j.jde.2011.09.006.

[14]

J. Geng and J. Wu, Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations, J. Math. Phys., 53 (2012), 102702. doi: 10.1063/1.4754822.

[15]

B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys., 307 (2011), 383-427. doi: 10.1007/s00220-011-1327-5.

[16]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, (Russian) Funktsional. Anal. iPrilozhen., 21 (1987), 22-37; English translation in: Funct. Anal. Appl., 21 (1987), 192-205.

[17]

S. B. Kuksin, Perturbations of conditionally periodic solutions of infinite-dimensional Hamiltonian systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41-63; English translation in: Math. USSR-Izv., 32 (1989), 39-62.

[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), 147-179.

[20]

S. B. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271. doi: 10.1007/PL00001476.

[21]

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

[22]

T. Kappeler and J. Pöschel, KdV and KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.

[23]

L. Jiao and Y. Wang, The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation, Commun. Pure Appl. Anal., 8 (2009), 1585-1606. doi: 10.3934/cpaa.2009.8.1585.

[24]

J. Liu and J. Si, Invariant tori for a derivative nonlinear Schrödinger equation with quasiperiodic forcing, J Math. Phys. , 56 (2015), 032702. doi: 10.1063/1.4916287.

[25]

J. Liu, Periodic and quasi-periodic solutions of a derivative nonlinear Schrödinger equation, Appl. Anal., 95 (2016), 801-825. doi: 10.1080/00036811.2015.1032942.

[26]

J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient, Comm. Pure. Appl. Math., 63 (2010), 1145-1172. doi: 10.1002/cpa.20314.

[27]

J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys., 307 (2011), 629-673. doi: 10.1007/s00220-011-1353-3.

[28]

J. Liu and X. Yuan, KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions, J. Differential Equations, 256 (2014), 1627-1652. doi: 10.1016/j.jde.2013.11.007.

[29]

L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasiperiodically forced perturbation, Discrete Contin. Dyn. Syst., 34 (2014), 689-707. doi: 10.3934/dcds.2014.34.689.

[30]

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

[31]

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

[32]

J. Rui and J. Si, Quasi-periodic solutions for quasi-periodically forced nonlinear Schrödinger equations with quasi-periodic inhomogeneous terms, Phys. D, 286/287 (2014), 1-31. doi: 10.1016/j.physd.2014.07.005.

[33]

M. B. Sevryuk, The reversible context 2 in KAM theory: the first steps, Regul. Chaotic Dyn., 16 (2011), 24-38. doi: 10.1134/S1560354710520035.

[34]

J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing, J. Differential Equations, 252 (2012), 5274-5360. doi: 10.1016/j.jde.2012.01.034.

[35]

Y. Wang and J. Si, A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term, Z. Angew. Math. Phys., 63 (2012), 189-190. doi: 10.1007/s00033-011-0172-x.

[36]

C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528.

[37]

X. Yuan, Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension, J. Differential Equations, 195 (2003), 230-242. doi: 10.1016/S0022-0396(03)00095-0.

[38]

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.

[39]

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.

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