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On the existence of full dimensional KAM torus for nonlinear Schrödinger equation
1. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
2. | College of Science, The Institute of Aeronautical Engineering and Technology, Binzhou University, Binzhou 256600, China |
3. | School of Mathematical Sciences, Peking University, Beijing 100871, China |
4. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
$ \begin{eqnarray} \sqrt{-1}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u = 0, \ x\in\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{eqnarray} $ |
$ V* $ |
$ \widehat{(V* u})_n = V_{n}\widehat{u}_n, V_n\in[-1, 1] $ |
$ f(x) $ |
$ 0\leq|\epsilon|\ll1 $ |
$ (V_n)_{n\in\mathbb{Z}} $ |
$ x $ |
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] |
P. Baldi, M. Berti and R. Montalto,
KAM for autonomous quasi-linear perturbations of KdV, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1589-1638.
doi: 10.1016/j.anihpc.2015.07.003. |
[3] |
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. |
[4] |
J. Bourgain,
Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[5] |
J. Bourgain,
Recent progress in quasi-periodic lattice Schrödinger operators and Hamiltonian partial differential equations, Russian Math. Surveys, 59 (2004), 231-246.
doi: 10.1070/RM2004v059n02ABEH000716. |
[6] |
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.![]() ![]() ![]() |
[7] |
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. |
[8] |
H. Z. Cong, J. J. Liu, Y. F. Shi and X. P. Yuan,
The stability of full dimensional KAM tori for nonlinear Schrödinger equation, J. Differential Equations, 264 (2018), 4504-4563.
doi: 10.1016/j.jde.2017.12.013. |
[9] |
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. |
[10] |
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. |
[11] |
R. Feola and M. Procesi,
Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations, 259 (2015), 3389-3447.
doi: 10.1016/j.jde.2015.04.025. |
[12] |
J. S. Geng,
Invariant tori of full dimension for a nonlinear Schrödinger equation, J. Differential Equations, 252 (2012), 1-34.
doi: 10.1016/j.jde.2011.09.006. |
[13] |
J. S. Geng and W. Hong,
Invariant tori of full dimension for second KdV equations with the external parameters, J. Dynam. Differential Equations, 29 (2017), 1325-1354.
doi: 10.1007/s10884-015-9505-3. |
[14] |
J. S. Geng and X. D. Xu,
Almost periodic solutions of one dimensional Schrödinger equation with the external parameters, J. Dynam. Differential Equations, 25 (2013), 435-450.
doi: 10.1007/s10884-013-9302-9. |
[15] |
T. Kappeler and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 45. Springer-Verlag, Berlin, 2003. |
[16] |
S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional. Anal. i Prilozhen., 21 (1987), 22–37, 95. |
[17] |
S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000. |
[18] |
S. B. Kuksin, Fifteen years of KAM for PDE., in Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 212 (2004), 237–258.
doi: 10.1090/trans2/212/12. |
[19] |
J. J. Liu and X. P. Yuan,
A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys., 307 (2011), 629-673.
doi: 10.1007/s00220-011-1353-3. |
[20] |
H. W. Niu and J. S. 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. |
[21] |
J. Pöschel,
Small divisors with spatial structure in infinite-dimensional Hamiltonian systems, Comm. Math. Phys., 127 (1990), 351-393.
doi: 10.1007/BF02096763. |
[22] |
J. Pöschel,
On the construction of almost periodic solutions for a nonlinear Schrödinger equation, Ergodic Theory Dynam. Systems, 22 (2002), 1537-1549.
doi: 10.1017/S0143385702001086. |
[23] |
C. E. Wayne,
Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528.
doi: 10.1007/BF02104499. |
[24] |
J. Wu and J. S. Geng,
Almost periodic solutions for a class of semilinear quantum harmonic oscillators, Discrete Contin. Dyn. Syst., 31 (2011), 997-1015.
doi: 10.3934/dcds.2011.31.997. |
[25] |
J. Zhang, M. 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] |
P. Baldi, M. Berti and R. Montalto,
KAM for autonomous quasi-linear perturbations of KdV, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1589-1638.
doi: 10.1016/j.anihpc.2015.07.003. |
[3] |
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. |
[4] |
J. Bourgain,
Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439.
doi: 10.2307/121001. |
[5] |
J. Bourgain,
Recent progress in quasi-periodic lattice Schrödinger operators and Hamiltonian partial differential equations, Russian Math. Surveys, 59 (2004), 231-246.
doi: 10.1070/RM2004v059n02ABEH000716. |
[6] |
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.![]() ![]() ![]() |
[7] |
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. |
[8] |
H. Z. Cong, J. J. Liu, Y. F. Shi and X. P. Yuan,
The stability of full dimensional KAM tori for nonlinear Schrödinger equation, J. Differential Equations, 264 (2018), 4504-4563.
doi: 10.1016/j.jde.2017.12.013. |
[9] |
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. |
[10] |
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. |
[11] |
R. Feola and M. Procesi,
Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations, 259 (2015), 3389-3447.
doi: 10.1016/j.jde.2015.04.025. |
[12] |
J. S. Geng,
Invariant tori of full dimension for a nonlinear Schrödinger equation, J. Differential Equations, 252 (2012), 1-34.
doi: 10.1016/j.jde.2011.09.006. |
[13] |
J. S. Geng and W. Hong,
Invariant tori of full dimension for second KdV equations with the external parameters, J. Dynam. Differential Equations, 29 (2017), 1325-1354.
doi: 10.1007/s10884-015-9505-3. |
[14] |
J. S. Geng and X. D. Xu,
Almost periodic solutions of one dimensional Schrödinger equation with the external parameters, J. Dynam. Differential Equations, 25 (2013), 435-450.
doi: 10.1007/s10884-013-9302-9. |
[15] |
T. Kappeler and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 45. Springer-Verlag, Berlin, 2003. |
[16] |
S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional. Anal. i Prilozhen., 21 (1987), 22–37, 95. |
[17] |
S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000. |
[18] |
S. B. Kuksin, Fifteen years of KAM for PDE., in Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 212 (2004), 237–258.
doi: 10.1090/trans2/212/12. |
[19] |
J. J. Liu and X. P. Yuan,
A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys., 307 (2011), 629-673.
doi: 10.1007/s00220-011-1353-3. |
[20] |
H. W. Niu and J. S. 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. |
[21] |
J. Pöschel,
Small divisors with spatial structure in infinite-dimensional Hamiltonian systems, Comm. Math. Phys., 127 (1990), 351-393.
doi: 10.1007/BF02096763. |
[22] |
J. Pöschel,
On the construction of almost periodic solutions for a nonlinear Schrödinger equation, Ergodic Theory Dynam. Systems, 22 (2002), 1537-1549.
doi: 10.1017/S0143385702001086. |
[23] |
C. E. Wayne,
Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528.
doi: 10.1007/BF02104499. |
[24] |
J. Wu and J. S. Geng,
Almost periodic solutions for a class of semilinear quantum harmonic oscillators, Discrete Contin. Dyn. Syst., 31 (2011), 997-1015.
doi: 10.3934/dcds.2011.31.997. |
[25] |
J. Zhang, M. 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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