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

Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system

Olivier Goubet 1, and  Marilena N. Poulou 2,

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.
Keywords: relaxation scheme., fractal dimension, global attractor, Klein-Gordon Schrödinger system.
Mathematics Subject Classification: Primary: 35Q55, 37L3.
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, J. Differential Equation, 83 (1990), 85-108. doi: 10.1016/0022-0396(90)90070-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

C. Besse, A relaxation scheme for nonlinear Schrödinger equations}, SIAM J. Num. Anal., 42 (2004), 934-952. doi: 10.1137/S0036142901396521.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schroinger equation, Differential Integral Equations, 23 (2010), 237-252.  Google Scholar

[10]

J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations, J. Math. Pures et Appli., 66 (1987), 273-319.  Google Scholar

[11]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[26]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

C. Besse, A relaxation scheme for nonlinear Schrödinger equations}, SIAM J. Num. Anal., 42 (2004), 934-952. doi: 10.1137/S0036142901396521.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schroinger equation, Differential Integral Equations, 23 (2010), 237-252.  Google Scholar

[10]

J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations, J. Math. Pures et Appli., 66 (1987), 273-319.  Google Scholar

[11]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[26]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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