Global attractor for a Klein-Gordon-Schrodinger type system
Marilena N. Poulou Nikolaos M. Stavrakakis
In this paper we prove the existence and uniqueness of solutions for the following evolution system of Klein-Gordon-Schrodinger type

$i\psi_t + k\psi_(xx) + i\alpha\psi$ = $\phi\psi + f(x)$,

$\phi_(tt)$ - $\phi_(xx) + \phi + \lambda\phi_t$ = -$Re\psi_x + g(x)$,

$\psi(x,0)=\psi_0(x), \phi(x,0)$ = $\phi_0, \phi_t(x,0)=\phi_1(x)$

$\phi(x,t)=\phi(x,t)=0$, $x\in\partial\Omega, t>0$

where $x \in \Omega, t > 0, k > 0, \alpha > 0, \lambda > 0, f(x)$ and $g(x)$ are the driving terms and $\Omega$ (bounded) $\subset \mathbb{R}$. Also we prove the continuous dependence of solutions of the system on the initial data as well as the existence of a global attractor.

keywords: Klein-Gordon-Schrodinger equation; Global Attractor; Absorbing set; Asymptotic Compactness; Uniqueness; Continuity.
Finite dimensionality of a Klein-Gordon-Schrödinger type system
Marilena N. Poulou Nikolaos M. Stavrakakis
In this paper we study the finite dimensionality of the global attractor for the following system of Klein-Gordon-Schrödinger type

$ 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.

keywords: Klein-Gordon-Schrödinger System; Global Attractor; Absorbing Set; Lyapunov Exponents; Hausdorff and Fractal Dimension
Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system
Olivier Goubet Marilena N. Poulou
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

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