November  2019, 39(11): 6599-6630. doi: 10.3934/dcds.2019287

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

* Corresponding author: Yuan Wu

Received  February 2019 Published  August 2019

Fund Project: H.C. is supported by the NNSFC (No. 11671066). L.M. is supported by the NNSFC (No. 11401041) and SPNSF (ZR2019MA062). Y.S. is supported by China Postdoctoral Science Foundation (No. 2018M641050). Y.W. is supported by NNSFC (No. 11790272 and No. 11421061).

In this paper, we study the following nonlinear Schrödinger equation
$ \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} $
where
$ V* $
is the Fourier multiplier defined by
$ \widehat{(V* u})_n = V_{n}\widehat{u}_n, V_n\in[-1, 1] $
and
$ f(x) $
is Gevrey smooth. It is shown that for
$ 0\leq|\epsilon|\ll1 $
, there is some
$ (V_n)_{n\in\mathbb{Z}} $
such that, the equation (1) admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain [7] and Cong-Liu-Shi-Yuan [8] to the case that the nonlinear perturbation depends explicitly on the space variable
$ x $
. The main difficulty here is the absence of zero momentum of the equation.
Citation: Hongzi Cong, Lufang Mi, Yunfeng Shi, Yuan Wu. On the existence of full dimensional KAM torus for nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6599-6630. doi: 10.3934/dcds.2019287
References:
[1]

P. BaldiM. 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. BaldiM. 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. CongJ. J. LiuY. 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. ZhangM. 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. BaldiM. 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. BaldiM. 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. CongJ. J. LiuY. 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. ZhangM. 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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