October  2020, 40(10): 5911-5928. doi: 10.3934/dcds.2020252

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

* Corresponding author: Jianjun Liu

Received  December 2019 Revised  April 2020 Published  June 2020

Fund Project: Meina Gao is supported by NNSFC 11971299. Jianjun Liu is supported by NNSFC 11671280, NNSFC 11822108, Fok Ying Tong Education Foundation 161002

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 [3] by Bambusi, Berti and Magistrelli. As applications, for nonlinear wave equation and nonlinear Schr$ \ddot{\mbox{o}} $dinger equation with periodic boundary conditions, quasi-periodic solutions of small amplitude and quasi-periodic solutions around plane wave are obtained respectively.

Citation: Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252
References:
[1]

P. BaldiM. BertiE. 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.  Google Scholar

[2]

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.  Google Scholar

[3]

D. BambusiM. 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.  Google Scholar

[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.  Google Scholar

[5]

M. BertiL. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Éc. Norm. Supér., 46 (2013), 301-373.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[12]

L. H. EliassonB. 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.  Google Scholar

[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.  Google Scholar

[14]

E. FaouL. 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.  Google Scholar

[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.  Google Scholar

[16]

J. GengX. 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.  Google Scholar

[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.  Google Scholar

[18]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21 (1987), 192-205.   Google Scholar

[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.  Google Scholar

[20]

S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math. 1556, Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0092243.  Google Scholar

[21] S. B Kuksin, Analysis of Hamiltonian PDEs, Oxford Univ. Press, Oxford, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.   Google Scholar

[28]

J. Pöschel, Quasi-periodic solutions for nonlinear wave equations, Comment. Math. Helv., 71 (1996), 269-296.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

P. BaldiM. BertiE. 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.  Google Scholar

[2]

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.  Google Scholar

[3]

D. BambusiM. 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.  Google Scholar

[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.  Google Scholar

[5]

M. BertiL. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Éc. Norm. Supér., 46 (2013), 301-373.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[12]

L. H. EliassonB. 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.  Google Scholar

[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.  Google Scholar

[14]

E. FaouL. 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.  Google Scholar

[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.  Google Scholar

[16]

J. GengX. 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.  Google Scholar

[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.  Google Scholar

[18]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21 (1987), 192-205.   Google Scholar

[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.  Google Scholar

[20]

S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math. 1556, Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0092243.  Google Scholar

[21] S. B Kuksin, Analysis of Hamiltonian PDEs, Oxford Univ. Press, Oxford, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.   Google Scholar

[28]

J. Pöschel, Quasi-periodic solutions for nonlinear wave equations, Comment. Math. Helv., 71 (1996), 269-296.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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