American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Analysis of a conservation law modeling a highly re-entrant manufacturing system
  • DCDS-B Home
  • This Issue
  • Next Article
    Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign
November  2010, 14(4): 1313-1335. doi: 10.3934/dcdsb.2010.14.1313

Supercritical surface waves generated by negative or oscillatory forcing

Jeongwhan Choi 1, , Tao Lin 2, , Shu-Ming Sun 2, and  Sungim Whang 3,

1. 

Department of Mathematics, Korea University, Seoul, South Korea
2. 

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, United States, United States
3. 

Department of Mathematics, Ajou University, Suwon, South Korea

Received  August 2009 Revised  March 2010 Published  August 2010

The paper studies forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small negative or oscillatory bump on a rigid flat bottom. Such wave motions are determined by a non-dimensional wave speed $F$, called Froude number, and $F=1$ is a critical value of $F$. If $F= 1+ \lambda \epsilon $ with a small parameter $\epsilon > 0$, then a forced Korteweg-de Vries (FKdV) equation can be derived to model the wave motion on the free surface. In this paper, the case $\lambda > 0$ (or $F> 1$, called supercritical case) is considered. The steady and unsteady solutions of the FKdV equation with a negative bump function independent of time are first studied both theoretically and numerically. It is shown that there are five steady solutions and only one of them, which exists for all $\lambda > 0$, is stable. Then, solutions of the FKdV equation with an oscillatory bump function posed on $R$ or a finite interval are considered. The corresponding linear problems are solved explicitly and the solutions are rigorously shown to be eventually periodic as time goes to infinity, while a similar result holds for the nonlinear problem posed on a finite interval with small initial data and forcing functions. The nonlinear solutions with zero initial data for any forcing functions in the real line $R$ or large forcing functions in a finite interval are obtained numerically. It is shown numerically that the solutions will become eventually periodic in time for a small forcing function. The behavior of the solutions becomes quite irregular as time goes to infinity, if the forcing function is large.
Keywords: Forced surface waves, negative forcing, oscillatory forcing..
Mathematics Subject Classification: Primary: 76B15; Secondary: 76B2.
Citation: Jeongwhan Choi, Tao Lin, Shu-Ming Sun, Sungim Whang. Supercritical surface waves generated by negative or oscillatory forcing. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1313-1335. doi: 10.3934/dcdsb.2010.14.1313
[1]

Roman Shvydkoy, Eitan Tadmor. Eulerian dynamics with a commutator forcing Ⅱ: Flocking. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5503-5520. doi: 10.3934/dcds.2017239
[2]

Martin Golubitsky, Claire Postlethwaite. Feed-forward networks, center manifolds, and forcing. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2913-2935. doi: 10.3934/dcds.2012.32.2913
[3]

T. Candan, R.S. Dahiya. Oscillation of mixed neutral differential equations with forcing term. Conference Publications, 2003, 2003 (Special) : 167-172. doi: 10.3934/proc.2003.2003.167
[4]

Hongjun Gao, Jinqiao Duan. Dynamics of the thermohaline circulation under wind forcing. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 205-219. doi: 10.3934/dcdsb.2002.2.205
[5]

Jinhuo Luo, Jin Wang, Hao Wang. Seasonal forcing and exponential threshold incidence in cholera dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2261-2290. doi: 10.3934/dcdsb.2017095
[6]

Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz, Huaiping Zhu. How seasonal forcing influences the complexity of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 785-807. doi: 10.3934/dcdsb.2018043
[7]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[8]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[9]

S. Rüdiger, J. Casademunt, L. Kramer. Kinks in stripe forming systems under traveling wave forcing. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1027-1042. doi: 10.3934/dcdsb.2005.5.1027
[10]

M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885
[11]

S.M. Booker, P.D. Smith, P. Brennan, R. Bullock. In-band disruption of a nonlinear circuit using optimal forcing functions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 221-242. doi: 10.3934/dcdsb.2002.2.221
[12]

Xuewei Ju, Desheng Li. Global synchronising behavior of evolution equations with exponentially growing nonautonomous forcing. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1921-1944. doi: 10.3934/cpaa.2018091
[13]

Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963
[14]

Amadeu Delshams, Vassili Gelfreich, Angel Jorba and Tere M. Seara. Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing. Electronic Research Announcements, 1997, 3: 1-10.

[15]

Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469
[16]

Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241
[17]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
[18]

Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267
[19]

Yahong Peng, Yaguang Wang. Reflection of highly oscillatory waves with continuous oscillatory spectra for semilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1293-1306. doi: 10.3934/dcds.2009.24.1293
[20]

Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences