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On the applicability of the poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models
KAM tori for quintic nonlinear schrödinger equations with given potential
1. | College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China |
2. | School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, 450001, China |
$ C^{\infty} $ |
$ \sqrt{-1}u_{t} = u_{xx}-V(x)u-|u|^4u $ |
References:
[1] |
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. |
[2] |
D. Bambusi and B. Grébert,
Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., 135 (2006), 507-567.
doi: 10.1215/S0012-7094-06-13534-2. |
[3] |
M. Berti, L. Biasco and M. Procesi,
KAM theory for the Hamiltonian derivative wave equations, Arch. Ration. Mech. Anal., 212 (2014), 905-955.
doi: 10.1007/s00205-014-0726-0. |
[4] |
J. Bourgain,
Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int. Math. Res. Not., 1994 (1994), 475-497.
doi: 10.1155/S1073792894000516. |
[5] |
J. Bourgain,
Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[6] |
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. |
[7] |
C. M. Cao and X. P. Yuan,
Quasi-peiodic solutions for perpertued generalized nonlinear vibrating string equation with singularities, Discrete Contin. Dyn. Syst., 37 (2017), 1867-1901.
doi: 10.3934/dcds.2017079. |
[8] |
L. Chierchia and J. G. You,
KAM tori for 1D nonlinear wave equation with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525.
doi: 10.1007/s002200050824. |
[9] |
L. J. Du and X. P. Yuan,
Invariant tori for nonlinear Schrödinger equations with a given potential, Dynamics of PDE, 3 (2006), 331-346.
doi: 10.4310/DPDE.2006.v3.n4.a4. |
[10] |
M. N. Gao and J. J. Liu,
Quasi-periodic solutions for 1D wave equation with higher order nonlinearity, J. Differential Equations, 252 (2012), 1466-1493.
doi: 10.1016/j.jde.2011.10.006. |
[11] |
T. Kappler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, Heidelberg, 2003.
doi: 10.1007/978-3-662-08054-2. |
[12] |
S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen., 21 (1987), 22–37, 95. |
[13] |
S. B. Kuksin,
Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR Izv., 32 (1989), 39-62.
doi: 10.1070/IM1989v032n01ABEH000733. |
[14] |
S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556. Springer-Verlag, Berlin, 1993.
doi: 10.1007/BFb0092243. |
[15] |
S. 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. |
[16] |
S. B. Kuksin,
A KAM theorem for equations of the Korteweg-de Vries type, Rev. Math-Math Phys., 10 (1998), 1-64.
|
[17] |
S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000.
![]() |
[18] |
Z. G. Liang and J. G. You,
Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity, SIAM J. Math. Anal., 36 (2005), 1965-1990.
doi: 10.1137/S0036141003435011. |
[19] |
J. J. Liu and X. P. Yuan,
Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient, Commun. Pure Appl. Math., 63 (2010), 1145-1172.
doi: 10.1002/cpa.20314. |
[20] |
J. J. Liu and X. P. 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. |
[21] |
J. J. Liu and X. P. Yuan,
KAM for the derivative nonliear Schrödinger equation with periodic boundary conditions, J. Differential Equations, 256 (2014), 1627-1652.
doi: 10.1016/j.jde.2013.11.007. |
[22] |
L. F. Mi,
Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential, J. Math. Anal. Appl., 390 (2012), 335-354.
doi: 10.1016/j.jmaa.2012.01.046. |
[23] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000.
![]() |
[24] |
J. Pöschel,
A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.
|
[25] |
J. Pöschel,
Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.
doi: 10.1007/BF02566420. |
[26] |
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part Ⅰ, Second Edition, Clarendon Press, Oxford, 1962.
![]() |
[27] |
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. |
[28] |
X. P. Yuan,
Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230 (2006), 213-274.
doi: 10.1016/j.jde.2005.12.012. |
[29] |
X. P. Yuan,
Invatiant tori of nonlinear wave equations with a given potential, Discrete Contin. Dyn. Syst., 16 (2006), 615-634.
doi: 10.3934/dcds.2006.16.615. |
[30] |
J. Zhang, M. N. Gao and X. P. Yuan,
KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1189-1228.
doi: 10.1088/0951-7715/24/4/010. |
show all references
References:
[1] |
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. |
[2] |
D. Bambusi and B. Grébert,
Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., 135 (2006), 507-567.
doi: 10.1215/S0012-7094-06-13534-2. |
[3] |
M. Berti, L. Biasco and M. Procesi,
KAM theory for the Hamiltonian derivative wave equations, Arch. Ration. Mech. Anal., 212 (2014), 905-955.
doi: 10.1007/s00205-014-0726-0. |
[4] |
J. Bourgain,
Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int. Math. Res. Not., 1994 (1994), 475-497.
doi: 10.1155/S1073792894000516. |
[5] |
J. Bourgain,
Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[6] |
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. |
[7] |
C. M. Cao and X. P. Yuan,
Quasi-peiodic solutions for perpertued generalized nonlinear vibrating string equation with singularities, Discrete Contin. Dyn. Syst., 37 (2017), 1867-1901.
doi: 10.3934/dcds.2017079. |
[8] |
L. Chierchia and J. G. You,
KAM tori for 1D nonlinear wave equation with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525.
doi: 10.1007/s002200050824. |
[9] |
L. J. Du and X. P. Yuan,
Invariant tori for nonlinear Schrödinger equations with a given potential, Dynamics of PDE, 3 (2006), 331-346.
doi: 10.4310/DPDE.2006.v3.n4.a4. |
[10] |
M. N. Gao and J. J. Liu,
Quasi-periodic solutions for 1D wave equation with higher order nonlinearity, J. Differential Equations, 252 (2012), 1466-1493.
doi: 10.1016/j.jde.2011.10.006. |
[11] |
T. Kappler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, Heidelberg, 2003.
doi: 10.1007/978-3-662-08054-2. |
[12] |
S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen., 21 (1987), 22–37, 95. |
[13] |
S. B. Kuksin,
Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR Izv., 32 (1989), 39-62.
doi: 10.1070/IM1989v032n01ABEH000733. |
[14] |
S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556. Springer-Verlag, Berlin, 1993.
doi: 10.1007/BFb0092243. |
[15] |
S. 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. |
[16] |
S. B. Kuksin,
A KAM theorem for equations of the Korteweg-de Vries type, Rev. Math-Math Phys., 10 (1998), 1-64.
|
[17] |
S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000.
![]() |
[18] |
Z. G. Liang and J. G. You,
Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity, SIAM J. Math. Anal., 36 (2005), 1965-1990.
doi: 10.1137/S0036141003435011. |
[19] |
J. J. Liu and X. P. Yuan,
Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient, Commun. Pure Appl. Math., 63 (2010), 1145-1172.
doi: 10.1002/cpa.20314. |
[20] |
J. J. Liu and X. P. 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. |
[21] |
J. J. Liu and X. P. Yuan,
KAM for the derivative nonliear Schrödinger equation with periodic boundary conditions, J. Differential Equations, 256 (2014), 1627-1652.
doi: 10.1016/j.jde.2013.11.007. |
[22] |
L. F. Mi,
Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential, J. Math. Anal. Appl., 390 (2012), 335-354.
doi: 10.1016/j.jmaa.2012.01.046. |
[23] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000.
![]() |
[24] |
J. Pöschel,
A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.
|
[25] |
J. Pöschel,
Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.
doi: 10.1007/BF02566420. |
[26] |
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part Ⅰ, Second Edition, Clarendon Press, Oxford, 1962.
![]() |
[27] |
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. |
[28] |
X. P. Yuan,
Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230 (2006), 213-274.
doi: 10.1016/j.jde.2005.12.012. |
[29] |
X. P. Yuan,
Invatiant tori of nonlinear wave equations with a given potential, Discrete Contin. Dyn. Syst., 16 (2006), 615-634.
doi: 10.3934/dcds.2006.16.615. |
[30] |
J. Zhang, M. N. Gao and X. P. Yuan,
KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1189-1228.
doi: 10.1088/0951-7715/24/4/010. |
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