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Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz–Krieger algebras
A degenerate KAM theorem for partial differential equations with periodic boundary conditions
1. | College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209, China |
2. | School of Mathematics, Sichuan University, Chengdu 610065, China |
In this paper, an infinite dimensional KAM theorem with double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with simple normal frequencies in [
References:
[1] |
P. Baldi, M. Berti, E. Haus and R. Montalto,
Time quasi-periodic gravity water waves in finite depth, Invent. Math., 214 (2018), 739-911.
doi: 10.1007/s00222-018-0812-2. |
[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, M. Berti and E. Magistrelli,
Degenerate KAM theory for partial differential equations, J. Differential Equations, 250 (2011), 3379-3397.
doi: 10.1016/j.jde.2010.11.002. |
[4] |
M. Berti and L. Biasco,
Branching of Cantor manifolds of elliptic tori and applications to PDEs, Commun. Math. Phys., 305 (2011), 741-796.
doi: 10.1007/s00220-011-1264-3. |
[5] |
M. Berti, L. Biasco and M. Procesi,
KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Éc. Norm. Supér., 46 (2013), 301-373.
|
[6] |
M. Berti and P. Bolle,
Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity, 25 (2012), 2579-2613.
doi: 10.1088/0951-7715/25/9/2579. |
[7] |
M. Berti and P. Bolle,
Quasi-periodic solutions with Sobolev regularity of NLS on $\Bbb T^d$ with a multiplicative potential, J. Eur. Math. Soc.(JEMS), 15 (2013), 229-286.
doi: 10.4171/JEMS/361. |
[8] |
J. Bourgain,
Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. of Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[9] |
J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005.
doi: 10.1515/9781400837144.![]() ![]() ![]() |
[10] |
L. Chierchia and J. You,
KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525.
doi: 10.1007/s002200050824. |
[11] |
W. Craig and C. E. Wayne,
Newton's method and periodic solutions of nonlinear wave equation, Comm. Pure Appl. Math., 46 (1993), 1409-1498.
doi: 10.1002/cpa.3160461102. |
[12] |
L. H. Eliasson, B. Grébert and S. B. Kuksin,
KAM for the nonlinear beam equation, Geom. Funct. Anal., 26 (2016), 1588-1715.
doi: 10.1007/s00039-016-0390-7. |
[13] |
L. H. Eliasson and S. B. Kuksin,
KAM for the nonlinear Schrödinger equation, Ann. of Math., 172 (2010), 371-435.
doi: 10.4007/annals.2010.172.371. |
[14] |
E. Faou, L. Gauckler and C. Lubich,
Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus, Comm. Partial Differential Equations, 38 (2013), 1123-1140.
doi: 10.1080/03605302.2013.785562. |
[15] |
J. Geng and J. You,
A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys., 262 (2006), 343-372.
doi: 10.1007/s00220-005-1497-0. |
[16] |
J. Geng, X. Xu and J. You,
An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., 226 (2011), 5361-5402.
doi: 10.1016/j.aim.2011.01.013. |
[17] |
B. Grébert and L. Thomann,
KAM for the quantum harmonic oscillator, Commun. Math. Phys., 307 (2011), 383-427.
doi: 10.1007/s00220-011-1327-5. |
[18] |
S. B. Kuksin,
Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21 (1987), 192-205.
|
[19] |
S. B. Kuksin,
Perturbations of conditionally periodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR-Izv., 32 (1989), 39-62.
doi: 10.1070/IM1989v032n01ABEH000733. |
[20] |
S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math. 1556, Springer-Verlag, Berlin, 1993.
doi: 10.1007/BFb0092243. |
[21] |
S. B Kuksin, Analysis of Hamiltonian PDEs, Oxford Univ. Press, Oxford, 2000.
![]() ![]() |
[22] |
S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149–179.
doi: 10.2307/2118656. |
[23] |
Z. Liang,
Quasi-periodic solutions for 1D Schrödinger equation with the nonlinearity $|u|^2pu$, J. Differenial Equations, 244 (2008), 2185-2225.
doi: 10.1016/j.jde.2008.02.015. |
[24] |
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. |
[25] |
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. |
[26] |
C. Procesi and M. Procesi,
A KAM algorithm for the completely resonant nonlinear Schrödinger equation, Adv. Math., 272 (2015), 399-470.
doi: 10.1016/j.aim.2014.12.004. |
[27] |
J. Pöschel,
A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.
|
[28] |
J. Pöschel,
Quasi-periodic solutions for nonlinear wave equations, Comment. Math. Helv., 71 (1996), 269-296.
|
[29] |
C. E. Wayne,
Periodic and quasi-periodic solutions of nonlinear wave equation via KAM theory, Commun. Math. Phys., 127 (1990), 479-528.
doi: 10.1007/BF02104499. |
[30] |
B. Wilson,
Sobolev stability of plane wave solutions to the nonlinear Schrödinger equation, Comm. Partial Differential Equations, 40 (2015), 1521-1542.
doi: 10.1080/03605302.2015.1030759. |
[31] |
X. 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. |
show all references
References:
[1] |
P. Baldi, M. Berti, E. Haus and R. Montalto,
Time quasi-periodic gravity water waves in finite depth, Invent. Math., 214 (2018), 739-911.
doi: 10.1007/s00222-018-0812-2. |
[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, M. Berti and E. Magistrelli,
Degenerate KAM theory for partial differential equations, J. Differential Equations, 250 (2011), 3379-3397.
doi: 10.1016/j.jde.2010.11.002. |
[4] |
M. Berti and L. Biasco,
Branching of Cantor manifolds of elliptic tori and applications to PDEs, Commun. Math. Phys., 305 (2011), 741-796.
doi: 10.1007/s00220-011-1264-3. |
[5] |
M. Berti, L. Biasco and M. Procesi,
KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Éc. Norm. Supér., 46 (2013), 301-373.
|
[6] |
M. Berti and P. Bolle,
Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity, 25 (2012), 2579-2613.
doi: 10.1088/0951-7715/25/9/2579. |
[7] |
M. Berti and P. Bolle,
Quasi-periodic solutions with Sobolev regularity of NLS on $\Bbb T^d$ with a multiplicative potential, J. Eur. Math. Soc.(JEMS), 15 (2013), 229-286.
doi: 10.4171/JEMS/361. |
[8] |
J. Bourgain,
Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. of Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[9] |
J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005.
doi: 10.1515/9781400837144.![]() ![]() ![]() |
[10] |
L. Chierchia and J. You,
KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525.
doi: 10.1007/s002200050824. |
[11] |
W. Craig and C. E. Wayne,
Newton's method and periodic solutions of nonlinear wave equation, Comm. Pure Appl. Math., 46 (1993), 1409-1498.
doi: 10.1002/cpa.3160461102. |
[12] |
L. H. Eliasson, B. Grébert and S. B. Kuksin,
KAM for the nonlinear beam equation, Geom. Funct. Anal., 26 (2016), 1588-1715.
doi: 10.1007/s00039-016-0390-7. |
[13] |
L. H. Eliasson and S. B. Kuksin,
KAM for the nonlinear Schrödinger equation, Ann. of Math., 172 (2010), 371-435.
doi: 10.4007/annals.2010.172.371. |
[14] |
E. Faou, L. Gauckler and C. Lubich,
Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus, Comm. Partial Differential Equations, 38 (2013), 1123-1140.
doi: 10.1080/03605302.2013.785562. |
[15] |
J. Geng and J. You,
A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys., 262 (2006), 343-372.
doi: 10.1007/s00220-005-1497-0. |
[16] |
J. Geng, X. Xu and J. You,
An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., 226 (2011), 5361-5402.
doi: 10.1016/j.aim.2011.01.013. |
[17] |
B. Grébert and L. Thomann,
KAM for the quantum harmonic oscillator, Commun. Math. Phys., 307 (2011), 383-427.
doi: 10.1007/s00220-011-1327-5. |
[18] |
S. B. Kuksin,
Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21 (1987), 192-205.
|
[19] |
S. B. Kuksin,
Perturbations of conditionally periodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR-Izv., 32 (1989), 39-62.
doi: 10.1070/IM1989v032n01ABEH000733. |
[20] |
S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math. 1556, Springer-Verlag, Berlin, 1993.
doi: 10.1007/BFb0092243. |
[21] |
S. B Kuksin, Analysis of Hamiltonian PDEs, Oxford Univ. Press, Oxford, 2000.
![]() ![]() |
[22] |
S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149–179.
doi: 10.2307/2118656. |
[23] |
Z. Liang,
Quasi-periodic solutions for 1D Schrödinger equation with the nonlinearity $|u|^2pu$, J. Differenial Equations, 244 (2008), 2185-2225.
doi: 10.1016/j.jde.2008.02.015. |
[24] |
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. |
[25] |
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. |
[26] |
C. Procesi and M. Procesi,
A KAM algorithm for the completely resonant nonlinear Schrödinger equation, Adv. Math., 272 (2015), 399-470.
doi: 10.1016/j.aim.2014.12.004. |
[27] |
J. Pöschel,
A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.
|
[28] |
J. Pöschel,
Quasi-periodic solutions for nonlinear wave equations, Comment. Math. Helv., 71 (1996), 269-296.
|
[29] |
C. E. Wayne,
Periodic and quasi-periodic solutions of nonlinear wave equation via KAM theory, Commun. Math. Phys., 127 (1990), 479-528.
doi: 10.1007/BF02104499. |
[30] |
B. Wilson,
Sobolev stability of plane wave solutions to the nonlinear Schrödinger equation, Comm. Partial Differential Equations, 40 (2015), 1521-1542.
doi: 10.1080/03605302.2015.1030759. |
[31] |
X. 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. |
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