Numerical study of a family of dissipative KdV equations

Previous Article
Resolvent estimates for a two-dimensional non-self-adjoint operator
CPAA Home
This Issue
Next Article
Incompressible type euler as scaling limit of compressible Euler-Maxwell equations

January 2013, 12(1): 519-546. doi: 10.3934/cpaa.2013.12.519

Numerical study of a family of dissipative KdV equations

Jean-Paul Chehab ^1, and Georges Sadaka ^1,

1.	LAMFA, UMR 6140, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens, France, France

Received April 2011 Revised December 2011 Published September 2012

Abstract
Related Papers

The weak damped and forced Korteweg-de Vries (KdV) equation on the 1d Torus have been analyzed by Ghidaglia[8, 9], Goubet[10, 11], Rosa and Cabral [3] where asymptotic regularization eects have been proven and observed numerically. In this work, we consider a family of dampings that can be even weaker, particularly it can dissipate very few the high frequencies. We give numerical evidences that point out dissipation of energy, regularization eect and the presence of special solutions that characterize a non trivial dynamics (steady states, time periodic solutions).

Keywords: Korteweg-de Vries equations, long time behavior, damping.

Mathematics Subject Classification: 35B10, 35B40, 35Q53, 65M12, 65M70.

Citation: Jean-Paul Chehab, Georges Sadaka. Numerical study of a family of dissipative KdV equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 519-546. doi: 10.3934/cpaa.2013.12.519

References:

[1]	M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems, Communications on Pure and Applied Analysis, 7 (2008), 211-227.
[2]	M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., ().
[3]	M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278. doi: 10.1016/j.physd.2004.01.023.
[4]	C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes, (French) [Multilevel schemes for waves equations], ESAIM Proc., 27, EDP Sci., (2009), 180-208. doi: 10.1051/proc/2009027.
[5]	J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings, Journal of Scientific Computing, 20 (2004), 159-189.
[6]	J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations, preprint, INRIA report, HAL, inria-00529227, (2011).
[7]	D. Dutykh, Modélisation mathématique des Tsunamis, (French) [Mathematical modeling of Tsunamis], Ph.D thesis, ENS Cachan, 2007.
[8]	J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eq., 74 (1988), 369-390. doi: 10.1016/0022-0396(88)90010-1.
[9]	J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations, J. Diff. Eq., 110 (1994), 356-359. doi: 10.1006/jdeq.1994.1071.
[10]	O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.
[11]	O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53. doi: 10.1006/jdeq.2001.4163.
[12]	D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications," SIAM Philadelphia, 1977. doi: 10.1115/1.3424477.
[13]	S. Mallat, "A Wavelet Tour of Signal Processing," Academic press, 1998.
[14]	A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics," Cambridge University Press, 2005.
[15]	L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, ().
[16]	E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping, Physics of fluids, 12 (1969), 2388-2394. doi: 10.1063/1.1692358.
[17]	E. Ott and R. N. Sudan, Damping of solitary waves, Physics of fluids, 13 (1970), 1432-1435. doi: 10.1063/1.1693097.
[18]	A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation, IMA J. Num. Anal., 20 (2000), 235-261. doi: 10.1093/imanum/20.2.235.
[19]	R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.
[20]	S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations, Funkcialaj Ekvacioj, 54 (2011), 119-138. doi: 10.1619/fesi.54.119.
[21]	S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations, Asymptot. Anal., 68 (2010), 155-186.

show all references

References:

[1]	M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems, Communications on Pure and Applied Analysis, 7 (2008), 211-227.
[2]	M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., ().
[3]	M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278. doi: 10.1016/j.physd.2004.01.023.
[4]	C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes, (French) [Multilevel schemes for waves equations], ESAIM Proc., 27, EDP Sci., (2009), 180-208. doi: 10.1051/proc/2009027.
[5]	J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings, Journal of Scientific Computing, 20 (2004), 159-189.
[6]	J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations, preprint, INRIA report, HAL, inria-00529227, (2011).
[7]	D. Dutykh, Modélisation mathématique des Tsunamis, (French) [Mathematical modeling of Tsunamis], Ph.D thesis, ENS Cachan, 2007.
[8]	J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eq., 74 (1988), 369-390. doi: 10.1016/0022-0396(88)90010-1.
[9]	J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations, J. Diff. Eq., 110 (1994), 356-359. doi: 10.1006/jdeq.1994.1071.
[10]	O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.
[11]	O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53. doi: 10.1006/jdeq.2001.4163.
[12]	D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications," SIAM Philadelphia, 1977. doi: 10.1115/1.3424477.
[13]	S. Mallat, "A Wavelet Tour of Signal Processing," Academic press, 1998.
[14]	A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics," Cambridge University Press, 2005.
[15]	L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, ().
[16]	E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping, Physics of fluids, 12 (1969), 2388-2394. doi: 10.1063/1.1692358.
[17]	E. Ott and R. N. Sudan, Damping of solitary waves, Physics of fluids, 13 (1970), 1432-1435. doi: 10.1063/1.1693097.
[18]	A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation, IMA J. Num. Anal., 20 (2000), 235-261. doi: 10.1093/imanum/20.2.235.
[19]	R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.
[20]	S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations, Funkcialaj Ekvacioj, 54 (2011), 119-138. doi: 10.1619/fesi.54.119.
[21]	S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations, Asymptot. Anal., 68 (2010), 155-186.

[1]	Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[2]	Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[3]	Brian Pigott. Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 389-418. doi: 10.3934/cpaa.2014.13.389
[4]	Belkacem Said-Houari. Long-time behavior of solutions of the generalized Korteweg--de Vries equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 245-252. doi: 10.3934/dcdsb.2016.21.245
[5]	Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control and Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
[6]	M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22
[7]	Julie Valein. On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021039
[8]	Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control and Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353
[9]	Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442
[10]	Olivier Goubet. Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 625-644. doi: 10.3934/dcds.2000.6.625
[11]	Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[12]	Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[13]	Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[14]	Ahmat Mahamat Taboye, Mohamed Laabissi. Exponential stabilization of a linear Korteweg-de Vries equation with input saturation. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021052
[15]	Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[16]	Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations and Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028
[17]	Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[18]	Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[19]	Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control and Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[20]	Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Jean-Paul Chehab Georges Sadaka
on Google Scholar
Jean-Paul Chehab Georges Sadaka

[Back to Top]