-
Previous Article
A dual-Petrov-Galerkin method for two integrable fifth-order KdV type equations
- DCDS Home
- This Issue
-
Next Article
Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean
Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability
| 1. | Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, United States |
| 2. | Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025 |
$ u_t +u_x +$uux$ +$uxxx$=0, \quad u(x,0)
=\phi (x), \qquad $ 0 < x < 1, t > 0, (*)
$ u(0,t) =h (t), \qquad
u(1,t)=0, \qquad u_x (1,t) =0, \quad t>0. $
It is shown that if the boundary forcing $h$ is periodic with small amplitude, then the small amplitude solution $u$ of (*) becomes eventually time-periodic. Viewing (*) (without the initial condition ) as an infinite-dimensional dynamical system in the Hilbert space $L^2(0,1)$, we also demonstrate that for a given periodic boundary forcing with small amplitude, the system (*) admits a (locally) unique limit cycle, or forced oscillation, which is locally exponentially stable. A list of open problems are included for the interested readers to conduct further investigations.
| [1] |
Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160 |
| [2] |
Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579 |
| [3] |
Lina Guo, Yulin Zhao. Existence of periodic waves for a perturbed quintic BBM equation. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4689-4703. doi: 10.3934/dcds.2020198 |
| [4] |
Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 47-66. doi: 10.3934/dcdss.2020003 |
| [5] |
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 |
| [6] |
Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185 |
| [7] |
S. Raynor, G. Staffilani. Low regularity stability of solitons for the KDV equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 277-296. doi: 10.3934/cpaa.2003.2.277 |
| [8] |
Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897 |
| [9] |
Xiongxiong Bao, Wan-Tong Li, Zhi-Cheng Wang. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Communications on Pure & Applied Analysis, 2020, 19 (1) : 253-277. doi: 10.3934/cpaa.2020014 |
| [10] |
Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure & Applied Analysis, 2004, 3 (2) : 301-318. doi: 10.3934/cpaa.2004.3.301 |
| [11] |
Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115 |
| [12] |
Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4579-4594. doi: 10.3934/dcdsb.2018177 |
| [13] |
Cyrine Fitouri, Alain Haraux. Boundedness and stability for the damped and forced single well Duffing equation. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 211-223. doi: 10.3934/dcds.2013.33.211 |
| [14] |
Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429 |
| [15] |
Xianbo Sun, Pei Yu. Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 965-987. doi: 10.3934/dcdsb.2018341 |
| [16] |
M. Burak Erdoğan, Nikolaos Tzirakis. Long time dynamics for forced and weakly damped KdV on the torus. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2669-2684. doi: 10.3934/cpaa.2013.12.2669 |
| [17] |
Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks & Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002 |
| [18] |
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 |
| [19] |
Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 |
| [20] |
Amin Esfahani. Remarks on a two dimensional BBM type equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1111-1127. doi: 10.3934/cpaa.2012.11.1111 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]