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Gevrey normal forms for nilpotent contact points of order two
Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation
1. | Department of Mathematics and Information Science, Binzhou University, Binzhou Shandong 256600, China |
2. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
References:
[1] |
M. J. Ablowitz and A. S. Fokas, The inverse scattering transform for the Benjamin-Ono equation-a pivot for multidimensional problems,, Stud. Appl. Math., 68 (1983), 1.
|
[2] |
D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods,, Comm. Math. Phys., 219 (2001), 465.
doi: 10.1007/s002200100426. |
[3] |
M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations,, Comm. Partial Differential Equations, 31 (2006), 959.
doi: 10.1080/03605300500358129. |
[4] |
Zh. Burgeĭn, Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs,, Russian Math. Surveys, 59 (2004), 231.
doi: 10.1070/RM2004v059n02ABEH000716. |
[5] |
M. Gao and J. Liu, Quasi-periodic solutions for derivative nonlinear Schrödinger equation,, Discrete Contin. Dyn. Syst., 32 (2012), 2101.
doi: 10.3934/dcds.2012.32.2101. |
[6] |
L. Jiao and Y. Wang, The construction of quasi-periocic solutions of quasi-periodic forced Schrödinger equation,, Commun. Pure Appl. Anal., 8 (2009), 1585.
doi: 10.3934/cpaa.2009.8.1585. |
[7] |
T. Kappeler and J. Pöschel, "KdV & KAM,", Ergebnisse der Mathematik und ihrer Grenzgebiete, 45 (2003).
|
[8] |
S. Klainerman, Long-time behaviour of solutions to nonliear wave equations,, in, (1984), 1209.
|
[9] |
S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation,, Ann. Math. (2), 143 (1996), 149.
doi: 10.2307/2118656. |
[10] |
S. B. Kuksin, On small-denominators equations with large variable coefficients,, Z. Angew. Math. Phys., 48 (1997), 262.
doi: 10.1007/PL00001476. |
[11] |
S. B. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford Lecture Series in Mathematics and its Applications, 19 (2000).
|
[12] |
S. B. Kuksin, Fifteen years of KAM in PDE,, in, 212 (2004), 237.
|
[13] |
P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations,, J. Math. Phys., 5 (1964), 611.
doi: 10.1063/1.1704154. |
[14] |
J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient,, Commun. Pure Appl. Math., 63 (2010), 1145.
doi: 10.1002/cpa.20314. |
[15] |
J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations,, Commun. Math. Phys., 307 (2011), 629.
doi: 10.1007/s00220-011-1353-3. |
[16] |
L. Mi, Quasi-periodic Solutions of derivative nonlinear Schrödinger equations with a given potential,, J. Math. Anal. Appl., 390 (2012), 335.
doi: 10.1016/j.jmaa.2012.01.046. |
[17] |
L. Molinet, Global well-posedness in the energy space for the Benjamin-Ono equation on the circle,, Math. Ann., 337 (2007), 353.
doi: 10.1007/s00208-006-0038-2. |
[18] |
L. Molinet, Global well-posedness in $L^2$ for the periodic Benjamin-Ono equation,, Amer. J. Math., 130 (2008), 635.
doi: 10.1353/ajm.0.0001. |
[19] |
J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations,, Comment. Math. Helv., 71 (1996), 269.
doi: 10.1007/BF02566420. |
[20] |
J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing,, J. Differential Equations, 252 (2012), 5274.
doi: 10.1016/j.jde.2012.01.034. |
[21] |
Y. Wang, Quasi-periodic solutions of a quasi-periodically forced nonlinear beam equation,, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2682.
doi: 10.1016/j.cnsns.2011.10.022. |
[22] |
J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differtial equations,, Nonliearity, 24 (2011), 1189.
doi: 10.1088/0951-7715/24/4/010. |
show all references
References:
[1] |
M. J. Ablowitz and A. S. Fokas, The inverse scattering transform for the Benjamin-Ono equation-a pivot for multidimensional problems,, Stud. Appl. Math., 68 (1983), 1.
|
[2] |
D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods,, Comm. Math. Phys., 219 (2001), 465.
doi: 10.1007/s002200100426. |
[3] |
M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations,, Comm. Partial Differential Equations, 31 (2006), 959.
doi: 10.1080/03605300500358129. |
[4] |
Zh. Burgeĭn, Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs,, Russian Math. Surveys, 59 (2004), 231.
doi: 10.1070/RM2004v059n02ABEH000716. |
[5] |
M. Gao and J. Liu, Quasi-periodic solutions for derivative nonlinear Schrödinger equation,, Discrete Contin. Dyn. Syst., 32 (2012), 2101.
doi: 10.3934/dcds.2012.32.2101. |
[6] |
L. Jiao and Y. Wang, The construction of quasi-periocic solutions of quasi-periodic forced Schrödinger equation,, Commun. Pure Appl. Anal., 8 (2009), 1585.
doi: 10.3934/cpaa.2009.8.1585. |
[7] |
T. Kappeler and J. Pöschel, "KdV & KAM,", Ergebnisse der Mathematik und ihrer Grenzgebiete, 45 (2003).
|
[8] |
S. Klainerman, Long-time behaviour of solutions to nonliear wave equations,, in, (1984), 1209.
|
[9] |
S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation,, Ann. Math. (2), 143 (1996), 149.
doi: 10.2307/2118656. |
[10] |
S. B. Kuksin, On small-denominators equations with large variable coefficients,, Z. Angew. Math. Phys., 48 (1997), 262.
doi: 10.1007/PL00001476. |
[11] |
S. B. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford Lecture Series in Mathematics and its Applications, 19 (2000).
|
[12] |
S. B. Kuksin, Fifteen years of KAM in PDE,, in, 212 (2004), 237.
|
[13] |
P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations,, J. Math. Phys., 5 (1964), 611.
doi: 10.1063/1.1704154. |
[14] |
J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient,, Commun. Pure Appl. Math., 63 (2010), 1145.
doi: 10.1002/cpa.20314. |
[15] |
J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations,, Commun. Math. Phys., 307 (2011), 629.
doi: 10.1007/s00220-011-1353-3. |
[16] |
L. Mi, Quasi-periodic Solutions of derivative nonlinear Schrödinger equations with a given potential,, J. Math. Anal. Appl., 390 (2012), 335.
doi: 10.1016/j.jmaa.2012.01.046. |
[17] |
L. Molinet, Global well-posedness in the energy space for the Benjamin-Ono equation on the circle,, Math. Ann., 337 (2007), 353.
doi: 10.1007/s00208-006-0038-2. |
[18] |
L. Molinet, Global well-posedness in $L^2$ for the periodic Benjamin-Ono equation,, Amer. J. Math., 130 (2008), 635.
doi: 10.1353/ajm.0.0001. |
[19] |
J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations,, Comment. Math. Helv., 71 (1996), 269.
doi: 10.1007/BF02566420. |
[20] |
J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing,, J. Differential Equations, 252 (2012), 5274.
doi: 10.1016/j.jde.2012.01.034. |
[21] |
Y. Wang, Quasi-periodic solutions of a quasi-periodically forced nonlinear beam equation,, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2682.
doi: 10.1016/j.cnsns.2011.10.022. |
[22] |
J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differtial equations,, Nonliearity, 24 (2011), 1189.
doi: 10.1088/0951-7715/24/4/010. |
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