
Previous Article
A dualPetrovGalerkin method for two integrable fifthorder 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 Kortewegde Vries equation on a bounded domain and their stability
1.  Department of Mathematics, University of Dayton, Dayton, OH 454692316, United States 
2.  Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 452210025 
$ u_t +u_x +$uu_{x}$ +$u_{xxx}$=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 timeperiodic. Viewing (*) (without the initial condition ) as an infinitedimensional 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, BingYu Zhang. Forced oscillation of viscous Burgers' equation with a timeperiodic force. Discrete and Continuous Dynamical Systems  B, 2021, 26 (2) : 12051221. doi: 10.3934/dcdsb.2020160 
[2] 
Seiji Ukai. Timeperiodic solutions of the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 579596. doi: 10.3934/dcds.2006.14.579 
[3] 
Lina Guo, Yulin Zhao. Existence of periodic waves for a perturbed quintic BBM equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 46894703. doi: 10.3934/dcds.2020198 
[4] 
Benjamin B. Kennedy. A periodic solution with nonsimple oscillation for an equation with statedependent delay and strictly monotonic negative feedback. Discrete and Continuous Dynamical Systems  S, 2020, 13 (1) : 4766. doi: 10.3934/dcdss.2020003 
[5] 
Yingte Sun, Xiaoping Yuan. Quasiperiodic solution of quasilinear fifthorder KdV equation. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 62416285. 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 and Continuous Dynamical Systems  B, 2020, 25 (1) : 321334. doi: 10.3934/dcdsb.2019185 
[7] 
Melek Jellouli. On the controllability of the BBM equation. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022002 
[8] 
S. Raynor, G. Staffilani. Low regularity stability of solitons for the KDV equation. Communications on Pure and Applied Analysis, 2003, 2 (3) : 277296. doi: 10.3934/cpaa.2003.2.277 
[9] 
JeanPaul Chehab, Pierre Garnier, Youcef Mammeri. Longtime behavior of solutions of a BBM equation with generalized damping. Discrete and Continuous Dynamical Systems  B, 2015, 20 (7) : 18971915. doi: 10.3934/dcdsb.2015.20.1897 
[10] 
Xiongxiong Bao, WanTong Li, ZhiCheng Wang. Uniqueness and stability of timeperiodic pyramidal fronts for a periodic competitiondiffusion system. Communications on Pure and Applied Analysis, 2020, 19 (1) : 253277. doi: 10.3934/cpaa.2020014 
[11] 
Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure and Applied Analysis, 2004, 3 (2) : 301318. doi: 10.3934/cpaa.2004.3.301 
[12] 
WeiJie Sheng, WanTong Li. Multidimensional stability of timeperiodic planar traveling fronts in bistable reactiondiffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 26812704. doi: 10.3934/dcds.2017115 
[13] 
Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete and Continuous Dynamical Systems  B, 2018, 23 (10) : 45794594. doi: 10.3934/dcdsb.2018177 
[14] 
Cyrine Fitouri, Alain Haraux. Boundedness and stability for the damped and forced single well Duffing equation. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 211223. doi: 10.3934/dcds.2013.33.211 
[15] 
Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2008, 7 (6) : 14291442. doi: 10.3934/cpaa.2008.7.1429 
[16] 
Xianbo Sun, Pei Yu. Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms. Discrete and Continuous Dynamical Systems  B, 2019, 24 (2) : 965987. doi: 10.3934/dcdsb.2018341 
[17] 
M. Burak Erdoğan, Nikolaos Tzirakis. Long time dynamics for forced and weakly damped KdV on the torus. Communications on Pure and Applied Analysis, 2013, 12 (6) : 26692684. doi: 10.3934/cpaa.2013.12.2669 
[18] 
Hirotada Honda. Globalintime solution and stability of KuramotoSakaguchi equation under nonlocal Coupling. Networks and Heterogeneous Media, 2017, 12 (1) : 2557. doi: 10.3934/nhm.2017002 
[19] 
Lei Jiao, Yiqian Wang. The construction of quasiperiodic solutions of quasiperiodic forced Schrödinger equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 15851606. doi: 10.3934/cpaa.2009.8.1585 
[20] 
Amin Esfahani. Remarks on a two dimensional BBM type equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 11111127. doi: 10.3934/cpaa.2012.11.1111 
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]