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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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[5]

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

[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.

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

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

[9]

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

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

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

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

[9]

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

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

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