American Institute of Mathematical Sciences

Advanced
April  2017, 37(4): 1867-1901. doi: 10.3934/dcds.2017079

Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities

Chengming Cao and  Xiaoping Yuan

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  October 2015 Revised  November 2016 Published  December 2016

Fund Project: Supported by NNSFC 11271076 and NNSFC 11121101.

The existence of 2-dimensional KAM tori is proved for the perturbed generalized nonlinear vibrating string equation with singularities $u_{tt}=((1-x^2)u_x)_x-mu-u^3$ subject to certain boundary conditions by means of infinite-dimensional KAM theory with the help of partial Birkhoff normal form, the characterization of the singular function space and the estimate of the integrals related to Legendre basis.

Keywords: KAM theory, quasi-periodic solutions, singular differential operator.
Mathematics Subject Classification: 37K55.
Citation: Chengming Cao, Xiaoping Yuan. Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1867-1901. doi: 10.3934/dcds.2017079
References:
[1]

M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Equations, 31 (2006), 959-985.  doi: 10.1080/03605300500358129.  Google Scholar

[2]

J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int. Math. Res. Not., 1994 (1994), 475-497.  doi: 10.1155/S1073792894000516.  Google Scholar

[3]

M. Gao and J. Liu, Quasi-periodic solutions for 1D wave equation with higher order nonlinearity, J. Differential Equations, 252 (2012), 1466-1493.  doi: 10.1016/j.jde.2011.10.006.  Google Scholar

[4]

B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys., 307 (2011), 383-427.  doi: 10.1007/s00220-011-1327-5.  Google Scholar

[5]

D. B. Henry, How to remember the Sobolev inequalities, Differential Equations, Springer Berlin Heidelberg, 957 (1982), 97-109.  Google Scholar

[6]

H. Y. Hsu, Certain integrals and infinite series involving ultra-spherical polynomials and Bessel functions, Duke Math. J., 4 (1938), 374-383.  doi: 10.1215/S0012-7094-38-00429-6.  Google Scholar

[7] S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Springer-Verlag, 1963.  doi: 10.1007/BFb0092243.  Google Scholar
[8]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, (Russian) Funktsional. Anal. i Prilozhen., 21 (1987), 22-37.   Google Scholar

[9]

S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR Izv., 32 (1989), 39-62.   Google Scholar

[10]

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

[11]

L. Nirenberg, An extended interpolation inequality, Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze, 20 (1966), 733-737.   Google Scholar

[12]

L. Nirenberg, On elliptic partial differential equations, IL Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali, 17 (2011), 1-48.  doi: 10.1007/978-3-642-10926-3_1.  Google Scholar

[13]

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

[14]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Super. Pisa, 23 (2006), 119-148.   Google Scholar

[15]

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.  Google Scholar

[16]

X. Yuan, Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation, Int. J. Math. Math. Sci., 2003 (2003), 1111-1136.  doi: 10.1155/S0161171203207092.  Google Scholar

[17]

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

[18]

X. Yuan, Invatiant tori of nonlinear wave equations with a given potential, Discrete Contin. Dyn. Syst., 16 (2006), 615-634.  doi: 10.3934/dcds.2006.16.615.  Google Scholar

show all references

References:
[1]

M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Equations, 31 (2006), 959-985.  doi: 10.1080/03605300500358129.  Google Scholar

[2]

J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int. Math. Res. Not., 1994 (1994), 475-497.  doi: 10.1155/S1073792894000516.  Google Scholar

[3]

M. Gao and J. Liu, Quasi-periodic solutions for 1D wave equation with higher order nonlinearity, J. Differential Equations, 252 (2012), 1466-1493.  doi: 10.1016/j.jde.2011.10.006.  Google Scholar

[4]

B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys., 307 (2011), 383-427.  doi: 10.1007/s00220-011-1327-5.  Google Scholar

[5]

D. B. Henry, How to remember the Sobolev inequalities, Differential Equations, Springer Berlin Heidelberg, 957 (1982), 97-109.  Google Scholar

[6]

H. Y. Hsu, Certain integrals and infinite series involving ultra-spherical polynomials and Bessel functions, Duke Math. J., 4 (1938), 374-383.  doi: 10.1215/S0012-7094-38-00429-6.  Google Scholar

[7] S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Springer-Verlag, 1963.  doi: 10.1007/BFb0092243.  Google Scholar
[8]

S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, (Russian) Funktsional. Anal. i Prilozhen., 21 (1987), 22-37.   Google Scholar

[9]

S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR Izv., 32 (1989), 39-62.   Google Scholar

[10]

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

[11]

L. Nirenberg, An extended interpolation inequality, Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze, 20 (1966), 733-737.   Google Scholar

[12]

L. Nirenberg, On elliptic partial differential equations, IL Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali, 17 (2011), 1-48.  doi: 10.1007/978-3-642-10926-3_1.  Google Scholar

[13]

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

[14]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Super. Pisa, 23 (2006), 119-148.   Google Scholar

[15]

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.  Google Scholar

[16]

X. Yuan, Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation, Int. J. Math. Math. Sci., 2003 (2003), 1111-1136.  doi: 10.1155/S0161171203207092.  Google Scholar

[17]

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

[18]

X. Yuan, Invatiant tori of nonlinear wave equations with a given potential, Discrete Contin. Dyn. Syst., 16 (2006), 615-634.  doi: 10.3934/dcds.2006.16.615.  Google Scholar

[1]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
[2]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[3]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[4]

Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104
[5]

Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585
[6]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171
[7]

Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615
[8]

Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019
[9]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[10]

Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839
[11]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020241
[12]

Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9
[13]

Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75
[14]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006
[15]

Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623
[16]

Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579
[17]

Koichiro Naito. Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 449-488. doi: 10.3934/dcds.2004.11.449
[18]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[19]

Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169
[20]

Xavier Blanc, Claude Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Networks & Heterogeneous Media, 2010, 5 (1) : 1-29. doi: 10.3934/nhm.2010.5.1

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (78)
  • HTML views (54)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences