Control Lie Algebras of Semi-Discretizations of the Schroedinger Equation

2007 ◽  
Author(s):  
Katherine A. Kime
Keyword(s):  
Lie Algebras ◽  
Space Variable ◽  
Single Point ◽  
Schroedinger Equation ◽  
Time Dependent ◽  
Full Width ◽  
One Dimensional ◽  
Space And Time ◽  
The One

We consider control of the one-dimensional Schroedinger equation via a time-dependent rectangular potential. We discretize the equation in the space variable, obtaining a system of ODEs in which the control is bilinear. We find Control Lie Algebras for several cases, including single point and full width potentials. We use full discretizations, in space and time, to examine the effect of the number of inputs.

Effect of the Spatial Extent of the Control in a Bilinear Control Problem for the Schroedinger Equation

2009 ◽  
Author(s):  
Katherine A. Kime
Keyword(s):  
Differential Geometry ◽  
Finite Difference ◽  
Lie Algebras ◽  
Grid Point ◽  
Schroedinger Equation ◽  
Time Varying ◽  
One Dimensional ◽  
Bilinear Control ◽  
The One ◽  
Bilinear Control Problem

We consider control of the one-dimensional Schroedinger equation through a time-varying potential. Using a finite difference semi-discretization, we consider increasing the extent of the potential from a single central grid-point in space to two or more gridpoints. With the differential geometry package in Maple 8, we compute and compare the corresponding Control Lie Algebras, identifying a trend in the number of elements which span the Control Lie Algebras.

A time-dependent model for high-pressure discharges in narrow ablative capillaries

1993 ◽  
Vol 50 (1) ◽  
pp. 51-70 ◽  
Author(s):  
D. Zoler ◽  
S. Cuperman ◽  
J. Ashkenazy ◽  
M. Caner ◽  
Z. Kaplan
Keyword(s):  
High Pressure ◽  
Equation Of State ◽  
Time Dependent ◽  
Dimensional Model ◽  
Plasma Sources ◽  
One Dimensional ◽  
Space And Time ◽  
Dependent Model ◽  
Energy Equations ◽  
Time Dependent Model

A time-dependent quasi-one-dimensional model is developed for studying high- pressure discharges in ablative capillaries used, for example, as plasma sources in electrothermal launchers. The main features of the model are (i) consideration of ablation effects in each of the continuity, momentum and energy equations; (ii) use of a non-ideal equation of state; and (iii) consideration of space- and time-dependent ionization.

scholarly journals On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 65-68 ◽  
Author(s):  
Bulent Kilic ◽  
Mustafa Inc ◽  
Dumitru Baleanu
Keyword(s):  
Soliton Solutions ◽  
First Integral ◽  
Optical Solitons ◽  
Time Dependent ◽  
Ansatz Method ◽  
One Dimensional ◽  
Optical Metamaterials ◽  
Schrödinger’S Equation ◽  
The One ◽  
Schrödinger's Equation

AbstractThis paper integrates dispersive optical solitons in special optical metamaterials with a time dependent coefficient. We obtained some optical solitons of the aforementioned equation. It is shown that the examined dependent coefficients are affected by the velocity of the wave. The first integral method (FIM) and ansatz method are applied to reach the optical soliton solutions of the one-dimensional nonlinear Schrödinger’s equation (NLSE) with time dependent coefficients.

RELAXATION AND TRANSIENTS IN A TIME-DEPENDENT LOGISTIC MAP

2002 ◽  
Vol 12 (07) ◽  
pp. 1667-1674 ◽  
Author(s):  
EDSON D. LEONEL ◽  
J. KAMPHORST LEAL DA SILVA ◽  
S. OLIFFSON KAMPHORST
Keyword(s):  
Control Parameter ◽  
Logistic Map ◽  
Period Doubling ◽  
Time Dependent ◽  
One Dimensional ◽  
Stability Period ◽  
The One ◽  
Exchange Of Stability ◽  
Small Periodic ◽  
Scaling Arguments

We study the one-dimensional logistic map with control parameter perturbed by a small periodic function. In the pure constant case, scaling arguments are used to obtain the exponents related to the relaxation of the trajectories at the exchange of stability, period-doubling and tangent bifurcations. In particular, we evaluate the exponent z which describes the divergence of the relaxation time τ near a bifurcation by the relation τ ~ | R - Rc |-z. Here, R is the control parameter and Rc is its value at the bifurcation. In the time-dependent case new attractors may appear leading to a different bifurcation diagram. Beside these new attractors, complex attractors also arise and are responsible for transients in many trajectories. We obtain, numerically, the exponents that characterize these transients and the relaxation of the trajectories.

scholarly journals On a direct method for the determination of an exact invariant for the time-dependent harmonic oscillator

1977 ◽  
Vol 20 (1) ◽  
pp. 97-105 ◽  
Author(s):  
P. G. L. Leach
Keyword(s):  
Direct Method ◽  
Dynamical Symmetry ◽  
Time Dependent ◽  
Symmetry Groups ◽  
New Approach ◽  
One Dimensional ◽  
The One ◽  
A Direct Method ◽  
Exact Invariant

AbstractAn exact invariant is found for the one-dimensional oscillator with equation of motion . The method used is that of linear canonical transformations with time-dependent coeffcients. This is a new approach to the problem and has the advantage of simplicity. When f(t) and g(t) are zero, the invariant is related to the well-known Lewis invariant. The significance of extension to higher dimension of these results is indicated, in particular for the existence of non-invariance dynamical symmetry groups.

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