July  2014, 13(4): 1525-1539. doi: 10.3934/cpaa.2014.13.1525

Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system

1. 

LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue St Leu, 80039, Amiens Cedex

2. 

Department of Mathematics, National Technical University, Zografou Campus 157 80, Athens, Greece

Received  July 2013 Revised  December 2013 Published  February 2014

We consider a semi-discrete in time relaxation scheme to discretize a damped forced nonlinear Klein-Gordon Schrödinger system. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a finite dimensional global attractor for this dynamical system.
Citation: Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525
References:
[1]

F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains,, \emph{J. Differential Equation, 83 (1990), 85. doi: 10.1016/0022-0396(90)90070-6.

[2]

M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete Schrödinger Equations and dissipative dynamical systems,, \emph{Comm. in Pure and Applied Analysis}, 7 (2008), 211.

[3]

M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrodinger system,, \emph{Differential Integral Equations}, 16 (2003), 573.

[4]

N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $R$,, \emph{Appl. Math. Lett.}, 12 (1999). doi: 10.1016/S0893-9659(98)00170-0.

[5]

J. Ball, Global attractors for damped semilinear wave equations,, \emph{Partial differential equations and applications, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31.

[6]

C. Besse, A relaxation scheme for nonlinear Schrödinger equations},, \emph{SIAM J. Num. Anal.}, 42 (2004), 934. doi: 10.1137/S0036142901396521.

[7]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, \emph{J. of Dyn. and Diff. Equ.}, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x.

[8]

M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation,, \emph{J. Comput. Phys.}, 44 (1981), 277. doi: 10.1016/0021-9991(81)90052-8.

[9]

E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schroinger equation,, \emph{Differential Integral Equations}, 23 (2010), 237.

[10]

J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations,, \emph{J. Math. Pures et Appli.}, 66 (1987), 273.

[11]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, \emph{Applicable Anal.}, 60 (1996), 99. doi: 10.1080/00036819608840420.

[12]

O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation,, \emph{Comm. in Pure and Applied Analysis}, 7 (2008), 1429. doi: 10.3934/cpaa.2008.7.1429.

[13]

B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrodinger equation in $R^3$,, \emph{J. Diff. Eq.}, 136 (1997), 356. doi: 10.1006/jdeq.1996.3242.

[14]

J. Hale, Asymptotic behavior of Dissipative Systems,, Math. surveys and Monographs, 25 (1988).

[15]

A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications, College de France Seminar, (1985).

[16]

N. Karachalios, M. Stavrakakis and P. Xanthopoulos, Parametric exponential energy decay for dissipative electron-ion plasma waves,, \emph{Zeitschrift foangewandte Mathematik und Physik ZAMP}, (2005), 218. doi: 10.1007/s00033-004-2095-2.

[17]

O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418.

[18]

P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 2 (1995), 357. doi: 10.1007/BF01261181.

[19]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains,, \emph{J. Diff. Eq.}, (2001), 281. doi: 10.1006/jdeq.2000.3827.

[20]

A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,, Handbook of Differential Equations, (). doi: 10.1016/S1874-5717(08)00003-0.

[21]

I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, \emph{Nonlinearity}, 11 (1998), 1369. doi: 10.1088/0951-7715/11/5/012.

[22]

M. N. Poulou and N. M. Stavrakakis, Global Attractor for a Klein-Gordon-Schrödinger Type System in all $R$,, \emph{Nonlinear Analysis: Theory, (): 2548. doi: 10.1016/j.na.2010.12.009.

[23]

M. N. Poulou and N. B. Zographopoulos, Global Attractor for a degenerate Klein - Gordon - Schrödinger Type System,, submitted., ().

[24]

G. Raugel, Global attractors in partial differential equations,, \emph{Handbook of dynamical systems}, (2002), 885. doi: 10.1016/S1874-575X(02)80038-8.

[25]

C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Applied Mathematical Sciences vol. 139, (1999).

[26]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).

[27]

B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equation,, \emph{J. of Math. Phy.}, 40 (1999). doi: 10.1063/1.532875.

[28]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrodinger equations and its applications to their attractors,, \emph{Physica D}, 88 (1995), 167. doi: 10.1016/0167-2789(95)00196-B.

[29]

Y. Yan, Attractors and dimensions for discretizations of a weakly damped Schrödinger equations and a sine-Gordon equation,, \emph{Nonlinear Anal.}, 20 (1993), 1417. doi: 10.1016/0362-546X(93)90168-R.

show all references

References:
[1]

F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains,, \emph{J. Differential Equation, 83 (1990), 85. doi: 10.1016/0022-0396(90)90070-6.

[2]

M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete Schrödinger Equations and dissipative dynamical systems,, \emph{Comm. in Pure and Applied Analysis}, 7 (2008), 211.

[3]

M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrodinger system,, \emph{Differential Integral Equations}, 16 (2003), 573.

[4]

N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $R$,, \emph{Appl. Math. Lett.}, 12 (1999). doi: 10.1016/S0893-9659(98)00170-0.

[5]

J. Ball, Global attractors for damped semilinear wave equations,, \emph{Partial differential equations and applications, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31.

[6]

C. Besse, A relaxation scheme for nonlinear Schrödinger equations},, \emph{SIAM J. Num. Anal.}, 42 (2004), 934. doi: 10.1137/S0036142901396521.

[7]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, \emph{J. of Dyn. and Diff. Equ.}, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x.

[8]

M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation,, \emph{J. Comput. Phys.}, 44 (1981), 277. doi: 10.1016/0021-9991(81)90052-8.

[9]

E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schroinger equation,, \emph{Differential Integral Equations}, 23 (2010), 237.

[10]

J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations,, \emph{J. Math. Pures et Appli.}, 66 (1987), 273.

[11]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, \emph{Applicable Anal.}, 60 (1996), 99. doi: 10.1080/00036819608840420.

[12]

O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation,, \emph{Comm. in Pure and Applied Analysis}, 7 (2008), 1429. doi: 10.3934/cpaa.2008.7.1429.

[13]

B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrodinger equation in $R^3$,, \emph{J. Diff. Eq.}, 136 (1997), 356. doi: 10.1006/jdeq.1996.3242.

[14]

J. Hale, Asymptotic behavior of Dissipative Systems,, Math. surveys and Monographs, 25 (1988).

[15]

A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications, College de France Seminar, (1985).

[16]

N. Karachalios, M. Stavrakakis and P. Xanthopoulos, Parametric exponential energy decay for dissipative electron-ion plasma waves,, \emph{Zeitschrift foangewandte Mathematik und Physik ZAMP}, (2005), 218. doi: 10.1007/s00033-004-2095-2.

[17]

O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418.

[18]

P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 2 (1995), 357. doi: 10.1007/BF01261181.

[19]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains,, \emph{J. Diff. Eq.}, (2001), 281. doi: 10.1006/jdeq.2000.3827.

[20]

A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,, Handbook of Differential Equations, (). doi: 10.1016/S1874-5717(08)00003-0.

[21]

I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, \emph{Nonlinearity}, 11 (1998), 1369. doi: 10.1088/0951-7715/11/5/012.

[22]

M. N. Poulou and N. M. Stavrakakis, Global Attractor for a Klein-Gordon-Schrödinger Type System in all $R$,, \emph{Nonlinear Analysis: Theory, (): 2548. doi: 10.1016/j.na.2010.12.009.

[23]

M. N. Poulou and N. B. Zographopoulos, Global Attractor for a degenerate Klein - Gordon - Schrödinger Type System,, submitted., ().

[24]

G. Raugel, Global attractors in partial differential equations,, \emph{Handbook of dynamical systems}, (2002), 885. doi: 10.1016/S1874-575X(02)80038-8.

[25]

C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Applied Mathematical Sciences vol. 139, (1999).

[26]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).

[27]

B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equation,, \emph{J. of Math. Phy.}, 40 (1999). doi: 10.1063/1.532875.

[28]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrodinger equations and its applications to their attractors,, \emph{Physica D}, 88 (1995), 167. doi: 10.1016/0167-2789(95)00196-B.

[29]

Y. Yan, Attractors and dimensions for discretizations of a weakly damped Schrödinger equations and a sine-Gordon equation,, \emph{Nonlinear Anal.}, 20 (1993), 1417. doi: 10.1016/0362-546X(93)90168-R.

[1]

Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695

[2]

Marilena N. Poulou, Nikolaos M. Stavrakakis. Global attractor for a Klein-Gordon-Schrodinger type system. Conference Publications, 2007, 2007 (Special) : 844-854. doi: 10.3934/proc.2007.2007.844

[3]

Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067

[4]

Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221

[5]

Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233

[6]

Marilena N. Poulou, Nikolaos M. Stavrakakis. Finite dimensionality of a Klein-Gordon-Schrödinger type system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 149-161. doi: 10.3934/dcdss.2009.2.149

[7]

E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156

[8]

Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413

[9]

Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040

[10]

Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081

[11]

Andrew Comech. Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2711-2755. doi: 10.3934/dcds.2013.33.2711

[12]

Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55

[13]

Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071

[14]

Pavlos Xanthopoulos, Georgios E. Zouraris. A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 239-263. doi: 10.3934/dcdsb.2008.10.239

[15]

Boling Guo, Yan Lv, Wei Wang. Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2795-2818. doi: 10.3934/dcds.2014.34.2795

[16]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903

[17]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679

[18]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076

[19]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359

[20]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485

2018 Impact Factor: 0.925

Metrics

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

Other articles
by authors

[Back to Top]