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 & Applied Analysis, 2017, 16 (1) : 25-68. doi: 10.3934/cpaa.2017002
References:
[1]

P. BaldiM. 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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, (Russian) Funktsional. Anal. iPrilozhen., 21 (1987), 22-37. Google Scholar

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S. B. Kuksin, Perturbations of conditionally periodic solutions of infinite-dimensional Hamiltonian systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41-63. Google Scholar

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

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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. Google Scholar

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

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

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

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[31]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

P. BaldiM. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, (Russian) Funktsional. Anal. iPrilozhen., 21 (1987), 22-37. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[31]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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