Almost periodic solutions for a class of semilinear quantum harmonic oscillators

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator,, preprint, ().
[10]	S. B. Kuksin, "Nearly Integrable Infinite Dimensional Hamiltonian Systems," Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993.
[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.
[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.
[13]	J. Pöschel, A KAM theorem for some nonlinear partial differential equations, Ann. Sc. Norm. sup. Pisa CI. Sci., 23 (1996), 119-148.
[14]	J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici., 71 (1993), 269-296.
[15]	J. Pöschel, On the construction of almost periodic solutions for a nonlinear Schrödinger equations, Ergod. Th. and Dynam. Syst., 22 (2002), 1537-1549.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator,, preprint, ().
[10]	S. B. Kuksin, "Nearly Integrable Infinite Dimensional Hamiltonian Systems," Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993.
[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.
[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.
[13]	J. Pöschel, A KAM theorem for some nonlinear partial differential equations, Ann. Sc. Norm. sup. Pisa CI. Sci., 23 (1996), 119-148.
[14]	J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici., 71 (1993), 269-296.
[15]	J. Pöschel, On the construction of almost periodic solutions for a nonlinear Schrödinger equations, Ergod. Th. and Dynam. Syst., 22 (2002), 1537-1549.
[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.

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