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September  2011, 31(3): 997-1015. doi: 10.3934/dcds.2011.31.997

Almost periodic solutions for a class of semilinear quantum harmonic oscillators

Jian Wu 1, and  Jiansheng Geng 1,

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China, China

Received  June 2010 Revised  October 2010 Published  August 2011

In this paper, we show that there are many almost periodic solutions corresponding to full dimensional invariant tori for the semilinear quantum harmonic oscillators with Hermite multiplier $${\rm i}{u}_{t}-u_{xx}+x^2u + M_\xi u+\varepsilon |u|^{2m}u=0,\quad u\in C^1(\Bbb R,L^2(\Bbb R)),$$ where $m \geq 1$ is an integer. The proof is based on an abstract infinite dimensional KAM theorem.
Keywords: KAM theory, Almost periodic solutions, Quantum harmonic oscillators..
Mathematics Subject Classification: Primary: 81Q55; Secondary: 81Q1.
Citation: Jian Wu, Jiansheng Geng. Almost periodic solutions for a class of semilinear quantum harmonic oscillators. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 997-1015. doi: 10.3934/dcds.2011.31.997
References:
[1]

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

[2]

J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 629-639. doi: 10.1007/BF01902055.  Google Scholar

[3]

J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, International Mathematics Research Notices, 1994, 475ff., approx. 21 pp.  Google Scholar

[4]

J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal., 6 (1996), 201-230. doi: 10.1007/BF02247885.  Google Scholar

[5]

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

[6]

W. Craig and C. E. 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

[7]

J. Geng and J. You, KAM tori of Hamiltonian perturbations of 1D linear beam equations, J. Math. Anal. Appl., 277 (2003), 104-121. doi: 10.1016/S0022-247X(02)00505-X.  Google Scholar

[8]

J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Commun. Math. Phys., 262 (2006), 343-372. doi: 10.1007/s00220-005-1497-0.  Google Scholar

[9]

B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator,, preprint, ().   Google Scholar

[10]

S. B. Kuksin, "Nearly Integrable Infinite Dimensional Hamiltonian Systems," Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993.  Google Scholar

[11]

S. B. 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

[12]

H. Niu and J. Geng, Almost periodic solutions for a class of higher dimensional beam equations, Nonlinearity, 20 (2007), 2499-2517. doi: 10.1088/0951-7715/20/11/003.  Google Scholar

[13]

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

[14]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici., 71 (1993), 269-296.  Google Scholar

[15]

J. Pöschel, On the construction of almost periodic solutions for a nonlinear Schrödinger equations, Ergod. Th. and Dynam. Syst., 22 (2002), 1537-1549. Google Scholar

[16]

K. Yajima and G. Zhang, Smoothing property for Schrödinger equations with potential superquadratic at infinity, Commun. Math. Phys., 221 (2001), 573-590. doi: 10.1007/s002200100483.  Google Scholar

show all references

References:
[1]

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

[2]

J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 629-639. doi: 10.1007/BF01902055.  Google Scholar

[3]

J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, International Mathematics Research Notices, 1994, 475ff., approx. 21 pp.  Google Scholar

[4]

J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal., 6 (1996), 201-230. doi: 10.1007/BF02247885.  Google Scholar

[5]

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

[6]

W. Craig and C. E. 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

[7]

J. Geng and J. You, KAM tori of Hamiltonian perturbations of 1D linear beam equations, J. Math. Anal. Appl., 277 (2003), 104-121. doi: 10.1016/S0022-247X(02)00505-X.  Google Scholar

[8]

J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Commun. Math. Phys., 262 (2006), 343-372. doi: 10.1007/s00220-005-1497-0.  Google Scholar

[9]

B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator,, preprint, ().   Google Scholar

[10]

S. B. Kuksin, "Nearly Integrable Infinite Dimensional Hamiltonian Systems," Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993.  Google Scholar

[11]

S. B. 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

[12]

H. Niu and J. Geng, Almost periodic solutions for a class of higher dimensional beam equations, Nonlinearity, 20 (2007), 2499-2517. doi: 10.1088/0951-7715/20/11/003.  Google Scholar

[13]

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

[14]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici., 71 (1993), 269-296.  Google Scholar

[15]

J. Pöschel, On the construction of almost periodic solutions for a nonlinear Schrödinger equations, Ergod. Th. and Dynam. Syst., 22 (2002), 1537-1549. Google Scholar

[16]

K. Yajima and G. Zhang, Smoothing property for Schrödinger equations with potential superquadratic at infinity, Commun. Math. Phys., 221 (2001), 573-590. doi: 10.1007/s002200100483.  Google Scholar

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