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Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities
School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
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.
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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