A degenerate KAM theorem for partial differential equations with periodic boundary conditions

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

A degenerate KAM theorem for partial differential equations with periodic boundary conditions

Meina Gao ^1, and Jianjun Liu ^2,,

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.

Keywords: Degenerate KAM theory, multiple normal frequencise.

Mathematics Subject Classification: Primary: 37K55; Secondary: 35Q55.

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

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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