American Institute of Mathematical Sciences

Advanced
January  2015, 35(1): 173-204. doi: 10.3934/dcds.2015.35.173

Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations

Meina Gao 1,

1. 

School of Science, Shanghai Second Polytechnic University, Shanghai 201209, China

Received  January 2014 Revised  April 2014 Published  August 2014

In this paper, we establish an estimate for the solutions of a small-divisor equation with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the derivative nonlinear Schrödinger equations subject to small Hamiltonian perturbations.
Keywords: multiple normal frequencise., KAM theory, derivative nonlinear Schrödinger equation.
Mathematics Subject Classification: Primary: 37K55; Secondary: 35Q5.
Citation: Meina Gao. Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 173-204. doi: 10.3934/dcds.2015.35.173
References:
[1]

P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type, Ann. Inst. H. Poincare (C) Anal. Non Linaire, 30 (2013), 33-77. doi: 10.1016/j.anihpc.2012.06.001.  Google Scholar

[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.  Google Scholar

[3]

D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480. doi: 10.1007/s002200100426.  Google Scholar

[4]

M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Ec. Norm. Super., 46 (2013), 301-373.  Google Scholar

[5]

N. N. Bogolyubov, Yu. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York, 1976.  Google Scholar

[6]

J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Res. Notices, 11 (1994), 475-497. doi: 10.1155/S1073792894000516.  Google Scholar

[7]

J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439. doi: 10.2307/121001.  Google Scholar

[8]

J. Bourgain, Harmoinc Analysis and Partial Differential Equations, University of Chicago Press, 1999.  Google Scholar

[9]

L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824.  Google Scholar

[10]

W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equation, Commun. Pure Appl. Math., 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102.  Google Scholar

[11]

L. Du and X. Yuan, Invariant tori of nonlinear Schrödinger equations with a given potential, Dynamics of PDE, 3 (2006), 331-346.  Google Scholar

[12]

L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371.  Google Scholar

[13]

M. Gao and J. Zhang, Small-divisor equation of higher order with large variable coefficient, Acta. Mathematica Sinica, English Series, 27 (2011), 2005-2032. doi: 10.1007/s10114-011-0064-1.  Google Scholar

[14]

T. Kappeler and J. Pöschel, KdV&KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.  Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin Heidelberg New York, 1980. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[16]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funkt. Anal. Prilozh., 21 (1987), 22-37; English translation in Funct. Anal. Appl., 21 (1987), 192-205.  Google Scholar

[17]

S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41-63; English translation in Math. USSR Izv., 32 (1989), 39-62.  Google Scholar

[18]

S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math., 1556, Springer-Verlag, New York, 1993.  Google Scholar

[19]

S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000.  Google Scholar

[20]

S. B. Kuksin and J. Pöschel, Invariant cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2), 143 (1996), 149-179. doi: 10.2307/2118656.  Google Scholar

[21]

P. Lancaster, Theory of Matrices, Academic Press LTD, New York and London, 1969.  Google Scholar

[22]

J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient, Comm. Pure Appl. Math., 63 (2010), 1145-1172. doi: 10.1002/cpa.20314.  Google Scholar

[23]

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

[24]

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

[25]

C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127 (1990), 479-528. doi: 10.1007/BF02104499.  Google Scholar

[26]

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

[27]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.  Google Scholar

[28]

J. You, Perturbations of lower dimensional tori for hamiltonian systems, J. Differential Equations, 152 (1999), 1-29. doi: 10.1006/jdeq.1998.3515.  Google Scholar

[29]

X. Yuan, A KAM theorem with appllications to partial differential equations of higher dimension, Commun. Math. Phys., 275 (2007), 97-137. doi: 10.1007/s00220-007-0287-2.  Google Scholar

[30]

J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1198-2118. doi: 10.1088/0951-7715/24/4/010.  Google Scholar

[31]

W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math., 120, Springer-Verlag, Berlin Heidelberg New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type, Ann. Inst. H. Poincare (C) Anal. Non Linaire, 30 (2013), 33-77. doi: 10.1016/j.anihpc.2012.06.001.  Google Scholar

[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.  Google Scholar

[3]

D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480. doi: 10.1007/s002200100426.  Google Scholar

[4]

M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Ec. Norm. Super., 46 (2013), 301-373.  Google Scholar

[5]

N. N. Bogolyubov, Yu. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York, 1976.  Google Scholar

[6]

J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Res. Notices, 11 (1994), 475-497. doi: 10.1155/S1073792894000516.  Google Scholar

[7]

J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439. doi: 10.2307/121001.  Google Scholar

[8]

J. Bourgain, Harmoinc Analysis and Partial Differential Equations, University of Chicago Press, 1999.  Google Scholar

[9]

L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824.  Google Scholar

[10]

W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equation, Commun. Pure Appl. Math., 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102.  Google Scholar

[11]

L. Du and X. Yuan, Invariant tori of nonlinear Schrödinger equations with a given potential, Dynamics of PDE, 3 (2006), 331-346.  Google Scholar

[12]

L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371.  Google Scholar

[13]

M. Gao and J. Zhang, Small-divisor equation of higher order with large variable coefficient, Acta. Mathematica Sinica, English Series, 27 (2011), 2005-2032. doi: 10.1007/s10114-011-0064-1.  Google Scholar

[14]

T. Kappeler and J. Pöschel, KdV&KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.  Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin Heidelberg New York, 1980. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[16]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funkt. Anal. Prilozh., 21 (1987), 22-37; English translation in Funct. Anal. Appl., 21 (1987), 192-205.  Google Scholar

[17]

S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41-63; English translation in Math. USSR Izv., 32 (1989), 39-62.  Google Scholar

[18]

S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math., 1556, Springer-Verlag, New York, 1993.  Google Scholar

[19]

S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000.  Google Scholar

[20]

S. B. Kuksin and J. Pöschel, Invariant cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2), 143 (1996), 149-179. doi: 10.2307/2118656.  Google Scholar

[21]

P. Lancaster, Theory of Matrices, Academic Press LTD, New York and London, 1969.  Google Scholar

[22]

J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient, Comm. Pure Appl. Math., 63 (2010), 1145-1172. doi: 10.1002/cpa.20314.  Google Scholar

[23]

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

[24]

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

[25]

C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127 (1990), 479-528. doi: 10.1007/BF02104499.  Google Scholar

[26]

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

[27]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.  Google Scholar

[28]

J. You, Perturbations of lower dimensional tori for hamiltonian systems, J. Differential Equations, 152 (1999), 1-29. doi: 10.1006/jdeq.1998.3515.  Google Scholar

[29]

X. Yuan, A KAM theorem with appllications to partial differential equations of higher dimension, Commun. Math. Phys., 275 (2007), 97-137. doi: 10.1007/s00220-007-0287-2.  Google Scholar

[30]

J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1198-2118. doi: 10.1088/0951-7715/24/4/010.  Google Scholar

[31]

W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math., 120, Springer-Verlag, Berlin Heidelberg New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

[1]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[2]

Hongzi Cong, Lufang Mi, Yunfeng Shi, Yuan Wu. On the existence of full dimensional KAM torus for nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6599-6630. doi: 10.3934/dcds.2019287
[3]

Razvan Mosincat, Haewon Yoon. Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 47-80. doi: 10.3934/dcds.2020003
[4]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[5]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93
[6]

Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102
[7]

Kazumasa Fujiwara, Tohru Ozawa. On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 275-280. doi: 10.3934/eect.2018013
[8]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
[9]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1447-1478. doi: 10.3934/cpaa.2021028
[10]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021028
[11]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[12]

Guanghua Shi, Dongfeng Yan. KAM tori for quintic nonlinear schrödinger equations with given potential. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2421-2439. doi: 10.3934/dcds.2020120
[13]

Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237
[14]

Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010
[15]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878
[16]

Fengshuang Gao, Yuxia Guo. Multiple solutions for a nonlinear Schrödinger systems. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1181-1204. doi: 10.3934/cpaa.2020055
[17]

Claudianor O. Alves, Chao Ji. Multiple positive solutions for a Schrödinger logarithmic equation. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2671-2685. doi: 10.3934/dcds.2020145
[18]

Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253
[19]

Nakao Hayashi, Pavel I. Naumkin, Patrick-Nicolas Pipolo. Smoothing effects for some derivative nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 685-695. doi: 10.3934/dcds.1999.5.685
[20]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences