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December  2013, 6(6): 1487-1506. doi: 10.3934/dcdss.2013.6.1487

On damping rates of dissipative KdV equations

Jean-Paul Chehab 1, and  Georges Sadaka 2,

1. 

LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, Pôle Scienti que, 33, rue Saint Leu, 80039 Amiens, France
2. 

LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens, France

Received  September 2012 Revised  September 2012 Published  April 2013

We consider here different models of dissipative Korteweg-de Vries (KdV) equations on the torus. Using a proper wave function $\Gamma$, we compare numerically the long time behavior effects of the damping models and we propose a hierarchy between these models. We also introduce a method based on the solution of an inverse problem to rebuild a posteriori the damping operator using only samples of the solution.
Keywords: long time behavior, damping rate, Korteweg-de Vries equations, inverse problem..
Mathematics Subject Classification: Primary: 35B40, 35Q53, 65M06, 65M32, 65M70; Secondary: 65L12, 15A2.
Citation: Jean-Paul Chehab, Georges Sadaka. On damping rates of dissipative KdV equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1487-1506. doi: 10.3934/dcdss.2013.6.1487
References:
[1]

J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4.

[2]

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.

[3]

J.-P. Chehab and G. Sadaka, Numerical study of a family of damped KdV equations, Communications on Pure and Applied Analysis,, 12 (2013), 519-546. doi: 10.3934/cpaa.2013.12.519.

[4]

M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with nonlocal viscous dispersive term, Discrete and Continuous Dynamical Systems, 27 (2010), 1473-1492. doi: 10.3934/dcds.2010.27.1473.

[5]

F. Dias and D. Dutykh, Viscous potentiel free-surface flows in a fluid layer of finite depth, C. R. Math. Acad. Sci. Paris, 345 (2007), 113-118. doi: 10.1016/j.crma.2007.06.007.

[6]

F. Dubois, Schemes available from:, \url{http://www.math.u-psud.fr/~fdubois/fractionnaire.html} [source Fortran]., (). 

[7]

F. Dubois, A. Galucio and N. Point, "Introduction à la Dérivation Fractionnaire. Théorie et Applications," (in French), Ref AF510, Techniques de l'ingénieur, April, 2010.

[8]

F. Dubois, J.-F. Deü and A. Galucio, The $G^\alpha$-scheme for approximation of fractional derivatives: Application to the dynamics of dissipative systems, J. Vib. Control, 14 (2008), 1597-1605. doi: 10.1177/1077546307087427.

[9]

S. Dumont and J.-B. Duval, Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity, International Journal of Numerical Analysis and Modeling, to appear, (2012).

[10]

D. Dutykh, "Modélisation Mathématique des Tsunamis," (French) [Mathematical modeling of Tsunamis], Ph.D thesis, ENS Cachan, 2007.

[11]

D. Dutykh, Visco-potential free-surface flows and long wave modelling, European Journal of Mechanics B Fluids, 28 (2009), 430-443. doi: 10.1016/j.euromechflu.2008.11.003.

[12]

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.

[13]

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.

[14]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.

[15]

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.

[16]

J. L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195 (2006), 6011-6045. doi: 10.1016/j.cma.2005.10.010.

[17]

J. Guerrero, M. Raydan and M. Rojas, A hybrid optimization method for large-scale non-negative full regularization in image restoration, Inverse Problems in Science ad Engineering, to appear, (2012). doi: 10.1080/17415977.2012.720684.

[18]

C. Hirsch, "Numerical Computation of Internal and External Flows. The Fundamentals of Computational Fluid Dynamics," Butterworth-Heinemann, 2007.

[19]

C. Jordan, "Calculus of Finite Differences," 3rd edition, Chelsea Publishing Co., New York, 1965.

[20]

C. Laurent, L. Rosier and B.-Y. Zhang, Control stabilization of the Korterweg-de Vries equation in a periodic domain, Comm. PDE, 35 (2010), 707-744. doi: 10.1080/03605300903585336.

[21]

S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42. doi: 10.1016/0021-9991(92)90324-R.

[22]

A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics," Second edition, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511755422.

[23]

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.

[24]

E. Ott and R. N. Sudan, Damping of solitary waves, Physics of Fluids, 13 (1970), 1432-1435. doi: 10.1063/1.1693097.

[25]

A. Pazoto and L. Rosier, Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line, DCDS-B, 14 (2010), 1511-1535. doi: 10.3934/dcdsb.2010.14.1511.

[26]

G. Sadaka, "Etude Mathématique et Numérique d'Équations d'Ondes Aquatiques Amorties," Thèse de Doctorat, Université de Picardie Jules Verne, November, 2011.

[27]

Lloyd N. Trefethen, "Spectral Methods in MATLAB," Software, Environments, and Tools, 10, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598.

show all references

References:
[1]

J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4.

[2]

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.

[3]

J.-P. Chehab and G. Sadaka, Numerical study of a family of damped KdV equations, Communications on Pure and Applied Analysis,, 12 (2013), 519-546. doi: 10.3934/cpaa.2013.12.519.

[4]

M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with nonlocal viscous dispersive term, Discrete and Continuous Dynamical Systems, 27 (2010), 1473-1492. doi: 10.3934/dcds.2010.27.1473.

[5]

F. Dias and D. Dutykh, Viscous potentiel free-surface flows in a fluid layer of finite depth, C. R. Math. Acad. Sci. Paris, 345 (2007), 113-118. doi: 10.1016/j.crma.2007.06.007.

[6]

F. Dubois, Schemes available from:, \url{http://www.math.u-psud.fr/~fdubois/fractionnaire.html} [source Fortran]., (). 

[7]

F. Dubois, A. Galucio and N. Point, "Introduction à la Dérivation Fractionnaire. Théorie et Applications," (in French), Ref AF510, Techniques de l'ingénieur, April, 2010.

[8]

F. Dubois, J.-F. Deü and A. Galucio, The $G^\alpha$-scheme for approximation of fractional derivatives: Application to the dynamics of dissipative systems, J. Vib. Control, 14 (2008), 1597-1605. doi: 10.1177/1077546307087427.

[9]

S. Dumont and J.-B. Duval, Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity, International Journal of Numerical Analysis and Modeling, to appear, (2012).

[10]

D. Dutykh, "Modélisation Mathématique des Tsunamis," (French) [Mathematical modeling of Tsunamis], Ph.D thesis, ENS Cachan, 2007.

[11]

D. Dutykh, Visco-potential free-surface flows and long wave modelling, European Journal of Mechanics B Fluids, 28 (2009), 430-443. doi: 10.1016/j.euromechflu.2008.11.003.

[12]

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.

[13]

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.

[14]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.

[15]

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.

[16]

J. L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195 (2006), 6011-6045. doi: 10.1016/j.cma.2005.10.010.

[17]

J. Guerrero, M. Raydan and M. Rojas, A hybrid optimization method for large-scale non-negative full regularization in image restoration, Inverse Problems in Science ad Engineering, to appear, (2012). doi: 10.1080/17415977.2012.720684.

[18]

C. Hirsch, "Numerical Computation of Internal and External Flows. The Fundamentals of Computational Fluid Dynamics," Butterworth-Heinemann, 2007.

[19]

C. Jordan, "Calculus of Finite Differences," 3rd edition, Chelsea Publishing Co., New York, 1965.

[20]

C. Laurent, L. Rosier and B.-Y. Zhang, Control stabilization of the Korterweg-de Vries equation in a periodic domain, Comm. PDE, 35 (2010), 707-744. doi: 10.1080/03605300903585336.

[21]

S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42. doi: 10.1016/0021-9991(92)90324-R.

[22]

A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics," Second edition, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511755422.

[23]

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.

[24]

E. Ott and R. N. Sudan, Damping of solitary waves, Physics of Fluids, 13 (1970), 1432-1435. doi: 10.1063/1.1693097.

[25]

A. Pazoto and L. Rosier, Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line, DCDS-B, 14 (2010), 1511-1535. doi: 10.3934/dcdsb.2010.14.1511.

[26]

G. Sadaka, "Etude Mathématique et Numérique d'Équations d'Ondes Aquatiques Amorties," Thèse de Doctorat, Université de Picardie Jules Verne, November, 2011.

[27]

Lloyd N. Trefethen, "Spectral Methods in MATLAB," Software, Environments, and Tools, 10, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598.

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