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Weighted Green functions of nondegenerate polynomial skew products on $\mathbb{C}^2$
Almost periodic solutions for a class of semilinear quantum harmonic oscillators
1. | Department of Mathematics, Nanjing University, Nanjing 210093, China, China |
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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