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Liouville theorem for an integral system on the upper half space
Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations
1. | School of Science, Shanghai Second Polytechnic University, Shanghai 201209, China |
References:
[1] |
P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type, Ann. Inst. H. Poincare (C) Anal. Non Linaire, 30 (2013), 33-77.
doi: 10.1016/j.anihpc.2012.06.001. |
[2] |
P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536.
doi: 10.1007/s00208-013-1001-7. |
[3] |
D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480.
doi: 10.1007/s002200100426. |
[4] |
M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Ec. Norm. Super., 46 (2013), 301-373. |
[5] |
N. N. Bogolyubov, Yu. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York, 1976. |
[6] |
J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Res. Notices, 11 (1994), 475-497.
doi: 10.1155/S1073792894000516. |
[7] |
J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[8] |
J. Bourgain, Harmoinc Analysis and Partial Differential Equations, University of Chicago Press, 1999. |
[9] |
L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525.
doi: 10.1007/s002200050824. |
[10] |
W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equation, Commun. Pure Appl. Math., 46 (1993), 1409-1498.
doi: 10.1002/cpa.3160461102. |
[11] |
L. Du and X. Yuan, Invariant tori of nonlinear Schrödinger equations with a given potential, Dynamics of PDE, 3 (2006), 331-346. |
[12] |
L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math., 172 (2010), 371-435.
doi: 10.4007/annals.2010.172.371. |
[13] |
M. Gao and J. Zhang, Small-divisor equation of higher order with large variable coefficient, Acta. Mathematica Sinica, English Series, 27 (2011), 2005-2032.
doi: 10.1007/s10114-011-0064-1. |
[14] |
T. Kappeler and J. Pöschel, KdV&KAM, Springer-Verlag, Berlin, 2003.
doi: 10.1007/978-3-662-08054-2. |
[15] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin Heidelberg New York, 1980.
doi: 10.1007/978-3-642-66282-9. |
[16] |
S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funkt. Anal. Prilozh., 21 (1987), 22-37; English translation in Funct. Anal. Appl., 21 (1987), 192-205. |
[17] |
S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41-63; English translation in Math. USSR Izv., 32 (1989), 39-62. |
[18] |
S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math., 1556, Springer-Verlag, New York, 1993. |
[19] |
S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000. |
[20] |
S. B. Kuksin and J. Pöschel, Invariant cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2), 143 (1996), 149-179.
doi: 10.2307/2118656. |
[21] |
P. Lancaster, Theory of Matrices, Academic Press LTD, New York and London, 1969. |
[22] |
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. |
[23] |
J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Commun. Math. Phys., 307 (2011), 629-673.
doi: 10.1007/s00220-011-1353-3. |
[24] |
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. |
[25] |
C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127 (1990), 479-528.
doi: 10.1007/BF02104499. |
[26] |
J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Sup. Pisa., 23 (1996), 119-148. |
[27] |
J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.
doi: 10.1007/BF02566420. |
[28] |
J. You, Perturbations of lower dimensional tori for hamiltonian systems, J. Differential Equations, 152 (1999), 1-29.
doi: 10.1006/jdeq.1998.3515. |
[29] |
X. Yuan, A KAM theorem with appllications to partial differential equations of higher dimension, Commun. Math. Phys., 275 (2007), 97-137.
doi: 10.1007/s00220-007-0287-2. |
[30] |
J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1198-2118.
doi: 10.1088/0951-7715/24/4/010. |
[31] |
W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math., 120, Springer-Verlag, Berlin Heidelberg New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
show all references
References:
[1] |
P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type, Ann. Inst. H. Poincare (C) Anal. Non Linaire, 30 (2013), 33-77.
doi: 10.1016/j.anihpc.2012.06.001. |
[2] |
P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536.
doi: 10.1007/s00208-013-1001-7. |
[3] |
D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480.
doi: 10.1007/s002200100426. |
[4] |
M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Ec. Norm. Super., 46 (2013), 301-373. |
[5] |
N. N. Bogolyubov, Yu. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York, 1976. |
[6] |
J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Res. Notices, 11 (1994), 475-497.
doi: 10.1155/S1073792894000516. |
[7] |
J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[8] |
J. Bourgain, Harmoinc Analysis and Partial Differential Equations, University of Chicago Press, 1999. |
[9] |
L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525.
doi: 10.1007/s002200050824. |
[10] |
W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equation, Commun. Pure Appl. Math., 46 (1993), 1409-1498.
doi: 10.1002/cpa.3160461102. |
[11] |
L. Du and X. Yuan, Invariant tori of nonlinear Schrödinger equations with a given potential, Dynamics of PDE, 3 (2006), 331-346. |
[12] |
L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math., 172 (2010), 371-435.
doi: 10.4007/annals.2010.172.371. |
[13] |
M. Gao and J. Zhang, Small-divisor equation of higher order with large variable coefficient, Acta. Mathematica Sinica, English Series, 27 (2011), 2005-2032.
doi: 10.1007/s10114-011-0064-1. |
[14] |
T. Kappeler and J. Pöschel, KdV&KAM, Springer-Verlag, Berlin, 2003.
doi: 10.1007/978-3-662-08054-2. |
[15] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin Heidelberg New York, 1980.
doi: 10.1007/978-3-642-66282-9. |
[16] |
S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funkt. Anal. Prilozh., 21 (1987), 22-37; English translation in Funct. Anal. Appl., 21 (1987), 192-205. |
[17] |
S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41-63; English translation in Math. USSR Izv., 32 (1989), 39-62. |
[18] |
S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math., 1556, Springer-Verlag, New York, 1993. |
[19] |
S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000. |
[20] |
S. B. Kuksin and J. Pöschel, Invariant cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2), 143 (1996), 149-179.
doi: 10.2307/2118656. |
[21] |
P. Lancaster, Theory of Matrices, Academic Press LTD, New York and London, 1969. |
[22] |
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. |
[23] |
J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Commun. Math. Phys., 307 (2011), 629-673.
doi: 10.1007/s00220-011-1353-3. |
[24] |
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. |
[25] |
C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127 (1990), 479-528.
doi: 10.1007/BF02104499. |
[26] |
J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Sup. Pisa., 23 (1996), 119-148. |
[27] |
J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.
doi: 10.1007/BF02566420. |
[28] |
J. You, Perturbations of lower dimensional tori for hamiltonian systems, J. Differential Equations, 152 (1999), 1-29.
doi: 10.1006/jdeq.1998.3515. |
[29] |
X. Yuan, A KAM theorem with appllications to partial differential equations of higher dimension, Commun. Math. Phys., 275 (2007), 97-137.
doi: 10.1007/s00220-007-0287-2. |
[30] |
J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1198-2118.
doi: 10.1088/0951-7715/24/4/010. |
[31] |
W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math., 120, Springer-Verlag, Berlin Heidelberg New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
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