Discrete Schrödinger equations and dissipative dynamical systems

Abstract

Abstract. We introduce a Crank-Nicolson scheme to study numerically the longtime behavior of solutions to a one dimensional damped forced nonlinear Schrödinger equation. We prove the existence of a smooth global attractor for these discretized equations. We also provide some numerical evidences of this asymptotical smoothing effect.

Long-time behavior of solutions of a BBM equation with generalized damping

“…Unfortunately, even if the used scheme is unconditionally stable, this method presents instabilities. The stability of the fixed-point iterate can be improved by applying extrapolation like technic as follows (see also [1]). The principle is to change the iterate v m+1 = Φ(u n , v m ) by j) denotes the j-th composition of Φ with itself.…”

Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs

“…For regular u, by the convergence of the serie, for N large enough, we have Z Y . • all the roots of p k are simple and alternates from p k to p k+1 ; they belong all in I: as a consequence p k oscillate more and more in I as k → +∞ hence the separation in frequencies, as illustrated in Figure (2).…”

“…At this point, we use ( 40) and obtain Unfortunately, in practice, this fixed point method converges only for very small values of ∆t. To enhance the stability region, and then to allow to take larger values of ∆t, we use the ∆ κ acceleration procedure introduced in [11] and applied in [1,2,19] for Allen-Cahn's, weakly damped Schrödinger and BBM equations respectively. In two words, the ∆ κ procedure consists in replacing the Picard iterates by…”

Global attractor for damped forced nonlinear logarithmic Schrödinger equations

“…The discrete counterpart of PDEs are numerical schemes. Considering semi-discrete in time schemes as discrete dynamical systems in infinite-dimensional Banach spaces lead to the issue of the existence of properties of global attractors for these discrete dynamical systems as in [2], [31], [23], [18]. To our knowledge, these issues are open for Log-NLS equations.…”