# American Institute of Mathematical Sciences

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 and 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, J. Differential Equation, 83 (1990), 85-108. 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, Comm. in Pure and Applied Analysis, 7 (2008), 211-227. [3] M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrodinger system, Differential Integral Equations, 16 (2003), 573-581. [4] N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $R$, Appl. Math. Lett., 12 (1999), 458. doi: 10.1016/S0893-9659(98)00170-0. [5] J. Ball, Global attractors for damped semilinear wave equations, Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [6] C. Besse, A relaxation scheme for nonlinear Schrödinger equations}, SIAM J. Num. Anal., 42 (2004), 934-952. doi: 10.1137/S0036142901396521. [7] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equ., 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x. [8] M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys., 44 (1981), 277-288. 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, Differential Integral Equations, 23 (2010), 237-252. [10] J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations, J. Math. Pures et Appli., 66 (1987), 273-319. [11] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420. [12] O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Comm. in Pure and Applied Analysis, 7 (2008), 1429-1442. doi: 10.3934/cpaa.2008.7.1429. [13] B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrodinger equation in $R^3$, J. Diff. Eq., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242. [14] J. Hale, Asymptotic behavior of Dissipative Systems, Math. surveys and Monographs, 25 (1988), AMS, Providence. [15] A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications College de France Seminar, vol 7, J. L. Lions and H. Brezis, Pitmann, London, 1985. [16] N. Karachalios, M. Stavrakakis and P. Xanthopoulos, Parametric exponential energy decay for dissipative electron-ion plasma waves, Zeitschrift foangewandte Mathematik und Physik ZAMP, (2005) Volume 56, Issue 2 , pp 218-238. doi: 10.1007/s00033-004-2095-2. [17] O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. [18] P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 357-369. doi: 10.1007/BF01261181. [19] K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Diff. Eq., 170 (2001), 281-316. 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, Nonlinearity, 11 (1998), 1369-1393. 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, Handbook of dynamical systems, Vol. 2, 885-982, North-Holland, Amsterdam, 2002. 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, Springer, 1999. [26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997. [27] B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equation, J. of Math. Phy., 40 (1999), 2445. 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, Physica D, 88 (1995), 167-175. 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, Nonlinear Anal., 20 (1993), 1417-1452. doi: 10.1016/0362-546X(93)90168-R.

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##### References:
 [1] F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equation, 83 (1990), 85-108. 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, Comm. in Pure and Applied Analysis, 7 (2008), 211-227. [3] M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrodinger system, Differential Integral Equations, 16 (2003), 573-581. [4] N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $R$, Appl. Math. Lett., 12 (1999), 458. doi: 10.1016/S0893-9659(98)00170-0. [5] J. Ball, Global attractors for damped semilinear wave equations, Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [6] C. Besse, A relaxation scheme for nonlinear Schrödinger equations}, SIAM J. Num. Anal., 42 (2004), 934-952. doi: 10.1137/S0036142901396521. [7] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equ., 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x. [8] M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys., 44 (1981), 277-288. 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, Differential Integral Equations, 23 (2010), 237-252. [10] J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations, J. Math. Pures et Appli., 66 (1987), 273-319. [11] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420. [12] O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Comm. in Pure and Applied Analysis, 7 (2008), 1429-1442. doi: 10.3934/cpaa.2008.7.1429. [13] B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrodinger equation in $R^3$, J. Diff. Eq., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242. [14] J. Hale, Asymptotic behavior of Dissipative Systems, Math. surveys and Monographs, 25 (1988), AMS, Providence. [15] A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications College de France Seminar, vol 7, J. L. Lions and H. Brezis, Pitmann, London, 1985. [16] N. Karachalios, M. Stavrakakis and P. Xanthopoulos, Parametric exponential energy decay for dissipative electron-ion plasma waves, Zeitschrift foangewandte Mathematik und Physik ZAMP, (2005) Volume 56, Issue 2 , pp 218-238. doi: 10.1007/s00033-004-2095-2. [17] O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. [18] P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 357-369. doi: 10.1007/BF01261181. [19] K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Diff. Eq., 170 (2001), 281-316. 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, Nonlinearity, 11 (1998), 1369-1393. 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, Handbook of dynamical systems, Vol. 2, 885-982, North-Holland, Amsterdam, 2002. 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, Springer, 1999. [26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997. [27] B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equation, J. of Math. Phy., 40 (1999), 2445. 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, Physica D, 88 (1995), 167-175. 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, Nonlinear Anal., 20 (1993), 1417-1452. doi: 10.1016/0362-546X(93)90168-R.
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