-
Previous Article
Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum
- DCDS-S Home
- This Issue
-
Next Article
Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions
Finite dimensionality of a Klein-Gordon-Schrödinger type system
| 1. | Department of Mathematics, National Technical University, Zografou Campus 157 80, Athens, Greece |
| 2. | Department of Mathematics, National Technical University, Zografou Campus 157 80, Athens, Hellas, Greece |
$ i\psi_t +\kappa \psi_{xx} +i\alpha\psi = \phi\psi+f,$
$ \phi_{tt}- \phi_{xx}+\phi+\lambda\phi_t = -Re \psi_{x}+g, $
$\psi (x,0)=\psi_0 (x), \phi(x,0) = \phi_0 (x), \phi_t (x,0)=\phi_1(x),$
$ \psi(x,t)= \phi(x,t)=0, x \in \partial \Omega,
t>0, $
where $x \in \Omega, t>0, \kappa > 0, \alpha >0, \lambda >0,$ $f$ and $g$ are driving terms and $\Omega$ is a bounded interval of R With the help of the Lyapunov exponents we give an estimate of the upper bound of its Hausdorff and Fractal dimension.
| [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] |
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 |
| [3] |
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 |
| [4] |
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 |
| [5] |
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 |
| [6] |
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 |
| [7] |
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 |
| [8] |
Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077 |
| [9] |
A. F. Almeida, M. M. Cavalcanti, J. P. Zanchetta. Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2039-2061. doi: 10.3934/cpaa.2018097 |
| [10] |
Caidi Zhao, Gang Xue, Grzegorz Łukaszewicz. Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4021-4044. doi: 10.3934/dcdsb.2018122 |
| [11] |
Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations & Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041 |
| [12] |
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 |
| [13] |
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 |
| [14] |
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 |
| [15] |
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 |
| [16] |
P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029 |
| [17] |
Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020216 |
| [18] |
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 |
| [19] |
P. Kaplický, Dalibor Pražák. Lyapunov exponents and the dimension of the attractor for 2d shear-thinning incompressible flow. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 961-974. doi: 10.3934/dcds.2008.20.961 |
| [20] |
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 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]