-
Previous Article
A Liouville type Theorem for an integral system
- CPAA Home
- This Issue
-
Next Article
Convergence to stationary solutions for a parabolic-hyperbolic phase-field system
Quasi-periodic solutions for 1D resonant beam equation
| 1. | School of Mathematical Sciences, Institute of Mathematics, Fudan University, Shanghai 200433, China |
| 2. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
$u_{t t} +u_{x x x x} +u^3=0,$
with hinged boundary conditions is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on infinite dimensional KAM theorem, partial normal form and scaling skills.
| [1] |
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 |
| [2] |
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 |
| [3] |
Bochao Chen, Yixian Gao. Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 921-944. doi: 10.3934/dcdsb.2021075 |
| [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] |
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 |
| [7] |
Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101 |
| [8] |
Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241 |
| [9] |
Michela Procesi. Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 541-552. doi: 10.3934/dcds.2005.13.541 |
| [10] |
Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467 |
| [11] |
Chengming Cao, Xiaoping Yuan. Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1867-1901. doi: 10.3934/dcds.2017079 |
| [12] |
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 |
| [13] |
Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4531-4543. doi: 10.3934/dcds.2021047 |
| [14] |
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 |
| [15] |
Jinjing Jiao, Guanghua Shi. Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5157-5180. doi: 10.3934/cpaa.2020231 |
| [16] |
Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615 |
| [17] |
Pietro Baldi. Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 883-903. doi: 10.3934/dcds.2006.15.883 |
| [18] |
Kai Tao. Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021162 |
| [19] |
Maurizio Garrione, Manuel Zamora. Periodic solutions of the Brillouin electron beam focusing equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 961-975. doi: 10.3934/cpaa.2014.13.961 |
| [20] |
Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]