scholarly journals Global dynamics of a non-local delayed differential equation in the half plane

2014 ◽  
Vol 13 (6) ◽  
pp. 2475-2492 ◽  
Author(s):  
Tao Wang
Keyword(s):  
Differential Equation ◽  
Half Plane ◽  
Global Dynamics ◽  
Delayed Differential Equation ◽  
Non Local

scholarly journals A CONVENIENT COORDINATIZATION OF SIEGEL–JACOBI DOMAINS

2012 ◽  
Vol 24 (10) ◽  
pp. 1250024 ◽  
Author(s):  
STEFAN BERCEANU
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Half Plane ◽  
Order Differential Equation ◽  
Quantum Evolution ◽  
Classical Motion ◽  
Matrix Riccati Equation ◽  
First Order ◽  
Upper Half Plane ◽  
Linear Differential

We determine the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi ball [Formula: see text] as the sum of the Kähler two-form on ℂn and the one on the Siegel ball [Formula: see text]. The classical motion and quantum evolution on [Formula: see text] determined by a hermitian linear Hamiltonian in the generators of the Jacobi group [Formula: see text] are described by a matrix Riccati equation on [Formula: see text] and a linear first-order differential equation in z ∈ ℂn, with coefficients depending also on [Formula: see text]. Hn denotes the (2n+1)-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on [Formula: see text]. When the transform FC : (η, W) → (z, W) is applied, the first-order differential equation in the variable [Formula: see text] becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel–Jacobi upper half plane [Formula: see text], where [Formula: see text] denotes the Siegel upper half plane.

scholarly journals Reduced ODE dynamics as formal relativistic asymptotics of a Peierls–Nabarro model

2014 ◽  
Vol 25 (4) ◽  
pp. 511-529
Author(s):  
H. IBRAHIM ◽  
R. MONNEAU
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Half Plane ◽  
Displacement Field ◽  
Equation Model ◽  
Linear Wave ◽  
Ordinary Differential Equation Model ◽  
Differential Equation Model ◽  
Linear Wave Equation ◽  
Total Displacement

In this paper, we consider a scalar Peierls--Nabarro model describing the motion of dislocations in the plane (x1,x2) along the linex2=0. Each dislocation can be seen as a phase transition and creates a scalar displacement field in the plane. This displacement field solves a simplified elasto-dynamics equation, which is simply a linear wave equation. The total displacement field creates a stress which makes move the dislocation itself. By symmetry, we can reduce the system to a wave equation in the half planex2>0 coupled with an equation for the dynamics of dislocations on the boundary of the half plane, i.e. onx2=0. Our goal is to understand the dynamics of well-separated dislocations in the limit when the distance between dislocations is very large, of order 1/ɛ. After rescaling, this corresponds to introduce a small parameter ɛ in the system. For the limit ɛ → 0, we are formally able to identify a reduced ordinary differential equation model describing the dynamics of relativistic dislocations if a certain conjecture is assumed to be true.

Sign in / Sign up

Export Citation Format

Share Document