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

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.
Citation: Meina Gao. Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations. Discrete & Continuous Dynamical Systems - A, 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.  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.  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.  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.   Google Scholar

[5]

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

[12]

L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation,, Ann. Math., 172 (2010), 371.  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, 27 (2011), 2005.  doi: 10.1007/s10114-011-0064-1.  Google Scholar

[14]

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

[15]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (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.   Google Scholar

[17]

S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems,, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41.   Google Scholar

[18]

S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems,, Lecture Notes in Math., (1556).   Google Scholar

[19]

S. B. Kuksin, Analysis of Hamiltonian PDEs,, Oxford University Press, (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.  doi: 10.2307/2118656.  Google Scholar

[21]

P. Lancaster, Theory of Matrices,, Academic Press LTD, (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.  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.  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.  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.  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.   Google Scholar

[27]

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

[28]

J. You, Perturbations of lower dimensional tori for hamiltonian systems,, J. Differential Equations, 152 (1999), 1.  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.  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.  doi: 10.1088/0951-7715/24/4/010.  Google Scholar

[31]

W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Math., (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.  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.  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.  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.   Google Scholar

[5]

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

[12]

L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation,, Ann. Math., 172 (2010), 371.  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, 27 (2011), 2005.  doi: 10.1007/s10114-011-0064-1.  Google Scholar

[14]

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

[15]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (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.   Google Scholar

[17]

S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems,, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41.   Google Scholar

[18]

S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems,, Lecture Notes in Math., (1556).   Google Scholar

[19]

S. B. Kuksin, Analysis of Hamiltonian PDEs,, Oxford University Press, (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.  doi: 10.2307/2118656.  Google Scholar

[21]

P. Lancaster, Theory of Matrices,, Academic Press LTD, (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.  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.  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.  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.  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.   Google Scholar

[27]

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

[28]

J. You, Perturbations of lower dimensional tori for hamiltonian systems,, J. Differential Equations, 152 (1999), 1.  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.  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.  doi: 10.1088/0951-7715/24/4/010.  Google Scholar

[31]

W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Math., (1989).  doi: 10.1007/978-1-4612-1015-3.  Google Scholar

[1]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[2]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[3]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[4]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[5]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[6]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[7]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[8]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[9]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[10]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[11]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[12]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[13]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[14]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[15]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[16]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044

[17]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[18]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[19]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[20]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]