REDUCED DYNAMICS FROM THE UNITARY GROUP TO SOME FLAG MANIFOLDS: INTERACTING MATRIX RICCATI EQUATIONS

In this paper we treat the time evolution of unitary elements in the N level system and consider the reduced dynamics from the unitary group U(N) to flag manifolds of the second type (in our terminology). Then we derive a set of differential equations of matrix Riccati types interacting with one another and present an important problem on a nonlinear superposition formula that the Riccati equation satisfies. Our result is a natural generalization of the paper Chaturvedi et al. [1] (arXiv: 0706.0964 [quant-ph]).

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THE NONLINEAR SUPERPOSITION PRINCIPLE AND THE WEI-NORMAN METHOD

International Journal of Modern Physics A ◽

10.1142/s0217751x98001694 ◽

1998 ◽

Vol 13 (21) ◽

pp. 3601-3627 ◽

Cited By ~ 26

Author(s):

J. F. CARIÑENA ◽

G. MARMO ◽

J. NASARRE

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Riccati Equation ◽

Order Differential Equation ◽

Superposition Principle ◽

Theoretical Perspective ◽

Nonlinear Superposition ◽

Theoretical Methods ◽

First Order ◽

Second Order Differential Equations

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrödinger equation.

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Density matrix and purity evolution in dissipative two-level systems: II. Relaxation

Physical Chemistry Chemical Physics ◽

10.1039/d0cp05528j ◽

2021 ◽

Author(s):

Sambarta Chatterjee ◽

Nancy Makri

Keyword(s):

Density Matrix ◽

Time Evolution ◽

Reduced Density Matrix ◽

Level System ◽

Two Level Systems ◽

Reduced Density ◽

Harmonic Bath

We investigate the time evolution of the reduced density matrix (RDM) and its purity in the dynamics of a two-level system coupled to a dissipative harmonic bath, when the system is initially placed in one of its eigenstates.

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Effective two-level reduction of N-level system dissipative quantum dynamics

Chemical Physics ◽

10.1016/0301-0104(90)87060-o ◽

1990 ◽

Vol 141 (2-3) ◽

pp. 249-259 ◽

Cited By ~ 3

Author(s):

Pavol Banacky

Keyword(s):

Quantum Dynamics ◽

Level System ◽

N Level ◽

Level Reduction

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Nonlinear superposition formula of the Novikov-Veselov equation

Journal of Physics A General Physics ◽

10.1088/0305-4470/27/4/026 ◽

1994 ◽

Vol 27 (4) ◽

pp. 1331-1338 ◽

Cited By ~ 38

Author(s):

Xing-Biao Hu

Keyword(s):

Nonlinear Superposition ◽

Nonlinear Superposition Formula

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Nonlinear superposition formula of the Boussinesq hierarchy

Acta Mathematicae Applicatae Sinica English Series ◽

10.1007/bf02009628 ◽

1993 ◽

Vol 9 (1) ◽

pp. 17-27 ◽

Cited By ~ 1

Author(s):

Xingbiao Hu ◽

Yong Li ◽

Qiming Liu

Keyword(s):

Nonlinear Superposition ◽

Nonlinear Superposition Formula

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Propagation of new dynamics of longitudinal bud equation among a magneto-electro-elastic round rod

Modern Physics Letters B ◽

10.1142/s0217984921503814 ◽

2021 ◽

Author(s):

Mostafa M. A. Khater ◽

Adil Jhangeer ◽

Hadi Rezazadeh ◽

Lanre Akinyemi ◽

M. Ali Akbar ◽

...

Keyword(s):

Differential Equations ◽

Riccati Equation ◽

Amorphous Silica ◽

Plasma Frequency ◽

Polarized Light ◽

Metal Films ◽

Mapping Method ◽

Longitudinal Waves ◽

Generalized Riccati Equation ◽

Amorphous Substrates

In this survey, structure solutions for the longitudinal suspense equation within a magneto-electro-elastic (MEE) circular judgment are extracted via the implementation of two one-of-a-kind techniques that are viewed as the close generalized technique in its field. This paper describes the dynamics of the longitudinal suspense within a MEE round rod. Nilsson and Lindau provided the visual proof of the arrival concerning longitudinal waves within thin metal films. They recommended removal of anomalies between the ports concerning emaciated ([Formula: see text] Å) Ag layers preserved on amorphous silica because of being mildly [Formula: see text]-polarized at frequency tier according to the brawny plasma frequency. Anomalies have been due to confusion over the longitudinal waves mirrored utilizing the two borders. The wavelength is connected according to this wave’s property, which is an awful lot smaller than the wave over light. The wave is outstanding only for altogether superfine films. However, metallic movies defended amorphous substrates that are intermittent within the forward levels of their evolution. This made researchers aware of the possibility on getting ready altogether few layers with a desire to discussing the promulgation about longitudinal waves up to expectation colorful each among reverberation or transport on [Formula: see text]-polarized light. The mated options via the use of generalized Riccati equation mapping method or generalized Kudryashov technique show the rule and effectiveness regarding its methods then its ability because of applying one kind of forms over nonlinear incomplete differential equations.

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CANONICAL FORM OF THE EVOLUTION OPERATOR OF A TIME-DEPENDENT HAMILTONIAN IN THE THREE LEVEL SYSTEM

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887811005312 ◽

2011 ◽

Vol 08 (03) ◽

pp. 647-655

Author(s):

KAZUYUKI FUJII

Keyword(s):

Differential Equations ◽

Canonical Form ◽

Evolution Operator ◽

Time Dependent ◽

Level System ◽

Riccati Differential Equations

In this paper we study the evolution operator of a time-dependent Hamiltonian in the three level system. The evolution operator is based on SU(3) and its dimension is eight, so we obtain three complex Riccati differential equations interacting with one another (which have been obtained by Fujii and Oike) and two real phase equations. This is a canonical form of the evolution operator.

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The nonlinear superposition theorem of Lie and Abel’s differential equations

Lettere al Nuovo Cimento ◽

10.1007/bf02789861 ◽

1983 ◽

Vol 38 (13) ◽

pp. 448-452 ◽

Cited By ~ 9

Author(s):

J. L. Reid ◽

G. L. Strobel

Keyword(s):

Differential Equations ◽

Nonlinear Superposition ◽

Superposition Theorem

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Propagation of bursting oscillations

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2009.0143 ◽

2009 ◽

Vol 367 (1908) ◽

pp. 4863-4875 ◽

Cited By ~ 8

Author(s):

Benjamin Ambrosio ◽

Jean-Pierre Françoise

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Numerical Simulations ◽

Time Evolution ◽

Global Attractor ◽

Reaction Diffusion ◽

Functional Space ◽

Excitable Cells ◽

Diffusion Type ◽

Bursting Oscillations

We investigate a system of partial differential equations of reaction–diffusion type which displays propagation of bursting oscillations. This system represents the time evolution of an assembly of cells constituted by a small nucleus of bursting cells near the origin immersed in the middle of excitable cells. We show that this system displays a global attractor in an appropriated functional space. Numerical simulations show the existence in this attractor of recurrent solutions which are waves propagating from the central source. The propagation seems possible if the excitability of the neighbouring cells is above some threshold.

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