Singular point analysis of the Gaudin equations

2008 ◽  
Vol 86 (6) ◽  
pp. 783-789 ◽  
Author(s):  
J R Schmidt
Keyword(s):  
Singular Point ◽  
Spin Model ◽  
Coupling Constants ◽  
Invariant Solutions ◽  
Painlevé Test ◽  
Rational Solutions ◽  
Systems Of Differential Equations ◽  
Point Analysis ◽  
All Solutions ◽  
Similarity Invariant

The Gaudin equations, a set of conditions of integrability imposed on the coupling constants of a lattice spin model, are solved by singular point analysis and the Painleve’ test. The Gaudin equations are transformed into systems of differential equations. The subset that are similarity-invariant have two, one, or zero constants of the motion corresponding to elliptic, trigonometric and (or) hyperbolic, and rational solutions, respectively. All solutions can be found at least formally by this technique. All similarity-invariant solutions are odd functions. There exist solutions with regular, even parts whose squares sum to a constant.PACS Nos.: 02.30.Ik, 02.40.Xx, 03.65.Fd

scholarly journals Non-polynomial Fourth Order Equations which Pass the Painlevé Test

2005 ◽  
Vol 60 (6) ◽  
pp. 387-400 ◽  
Author(s):  
Fahd Jrad ◽  
Uğurhan Muğan
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Painlevé Test ◽  
Point Analysis ◽  
Pacs 02.30 ◽  
Fourth Order Equations ◽  
Polynomial Class

The singular point analysis of fourth order ordinary differential equations in the non-polynomial class are presented. Some new fourth order ordinary differential equations which pass the Painlevé test as well as the known ones are found. -PACS: 02.30.Hq, 02.30.Ik, 02.30.Gp

scholarly journals Non-polynomial Third Order Equations which Pass the Painlevé Test

2004 ◽  
Vol 59 (3) ◽  
pp. 163-180 ◽  
Author(s):  
Uǧurhan Muǧan ◽  
Fahd Jrad
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Painlevé Test ◽  
Third Order ◽  
Point Analysis ◽  
Polynomial Class

The singular point analysis of third-order ordinary differential equations in the non-polynomial class is presented. Some new third order ordinary differential equations which pass the Painlevé test, as well as the known ones are found.

scholarly journals Existence results for differential equations with reflection of the argument

1994 ◽  
Vol 57 (2) ◽  
pp. 237-260 ◽  
Author(s):  
Donal O'Regan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Existence Results ◽  
Second Order ◽  
Systems Of Differential Equations ◽  
Nonlinear Alternative ◽  
Point Analysis ◽  
Second Order Equations

AbstractExistence principles are given for systems of differential equations with reflection of the argument. These are derived using fixed point analysis, specifically the Nonlinear Alternative. Then existence results are deduced for certain classes of first and second order equations with reflection of the argument.

Integrability of dynamical systems and the singular-point analysis

1985 ◽  
Vol 15 (6) ◽  
pp. 637-666 ◽  
Author(s):  
W. -H. Steeb ◽  
M. Kloke ◽  
B. M. Spieker ◽  
A. Kunick
Keyword(s):  
Dynamical Systems ◽  
Singular Point ◽  
Point Analysis

Hamiltonian systems with three degrees of freedom, singular-point analysis, and chaotic behavior

1986 ◽  
Vol 33 (3) ◽  
pp. 2131-2133 ◽  
Author(s):  
W.-H. Steeb ◽  
J. A. Louw ◽  
P. G. L. Leach ◽  
F. M. Mahomed
Keyword(s):  
Singular Point ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Point Analysis ◽  
Three Degrees Of Freedom

Physical aspects of the relaxation model in two-phase flow

1990 ◽  
Vol 428 (1875) ◽  
pp. 379-397 ◽  
Keyword(s):  
Phase Space ◽  
Singular Point ◽  
Shock Waves ◽  
Saddle Points ◽  
Singular Points ◽  
Relaxation Model ◽  
Two Phase ◽  
Point Patterns ◽  
All Solutions ◽  
Topological Pattern

The paper explores the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows. To this end, a rigorous mathematical analysis highlights the similarities and differences between this and the homogeneous equilibrium model (HEM) emphasizing the physical and qualitative aspects of the problem. Special attention is placed on a study of dispersion, characteristics, choking and shock waves. The most essential features are discovered with reference to the appropriate and convenient phase space Ω for HRM, which consists of pressure P , enthalpy h , dryness fraction x , velocity w , and length coordinate z . The geometric properties of the phase space Ω enable us to sketch the topological pattern of all solutions of the model. The study of choking is intimately connected with the occurrence of singular points of the set of simultaneous first-order differential equations of the model. The very powerful centre manifold theorem allows us to reduce the study of singular points to a two-dimensional plane Π , which is tangent to the solutions at a singular point, and so to demonstrate that only three singular-point patterns can appear (excepting degenerate cases), namely saddle points, nodal points and spiral points. The analysis reveals the existence of two limiting velocities of wave propagation, the frozen velocity a f and the equilibrium velocity a e . The critical velocity of choking is the frozen speed of sound. The analysis proves unequivocally that transition from ω < a f to w > a f can take place only via a singular point. Such a condition can also be attained at the end of a channel. The paper concludes with a short discussion of normal, fully dispersed and partly dispersed shock waves.

Singular point analysis, resonances and Yoshida's theorem

1987 ◽  
Vol 20 (12) ◽  
pp. 4027-4030 ◽  
Author(s):  
W -H Steeb ◽  
J A Louw ◽  
M F Maritz
Keyword(s):  
Singular Point ◽  
Point Analysis
Sign in / Sign up

Export Citation Format

Share Document