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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 & 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

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