scholarly journals The Adler-van Moerbeke integrable case. Visualization of bifurcations of Liouville tori

2017 ◽  
Vol 27 (4) ◽  
pp. 532-539
Author(s):  
S.V. Sokolov ◽  
Keyword(s):  
Integrable Case ◽  
Liouville Tori

scholarly journals Bifurcations of Liouville tori in a system of two vortices of positive intensity in a Bose-Einstein condensate

2019 ◽  
Vol 485 (6) ◽  
pp. 670-675
Author(s):  
P. E. Ryabov
Keyword(s):  
Hamiltonian System ◽  
Degrees Of Freedom ◽  
Integrable Hamiltonian System ◽  
Einstein Condensate ◽  
Integrable Case ◽  
Vortex Pairs ◽  
Bose Einstein Condensate ◽  
Liouville Tori ◽  
Bose Einstein ◽  
Liouville Integrable

In this paper we consider a completely Liouville integrable Hamiltonian system with two degrees of freedom, which describes the dynamics of two vortex filaments in a Bose-Einstein condensate enclosed in a harmonic trap. For vortex pairs of positive intensity detected bifurcation of three Liouville tori into one. Such bifurcation was found in the integrable case of Goryachev-Chaplygin-Sretensky in the dynamics of a rigid body. For the integrable perturbation of the physical parameter of the intensity ratio, identified bifurcation proved to be unstable, which led to bifurcations of the type of two tori into one and vice versa.

Bifurcations of Liouville tori of an integrable case of swinging Atwood's machine

1995 ◽  
Vol 110 (9) ◽  
pp. 1111-1121 ◽  
Author(s):  
A. Ouazzani-T.H. ◽  
M. Ouazzani-Jamil
Keyword(s):  
Integrable Case ◽  
Liouville Tori

scholarly journals Quasi-periodic solutions of the coupled nonlinear Schrödinger equations

1995 ◽  
Vol 451 (1943) ◽  
pp. 685-700 ◽  
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Schrödinger Equations ◽  
Integrable Case ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Fibre Optics ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Coupled Nonlinear Schrodinger Equations

We consider travelling periodic and quasi-periodic wave solutions of a set of coupled nonlinear Schrödinger equations. In fibre optics these equations can be used to model single mode fibres under the action of cross-phase modulation, with weak birefringence. The problem is reduced to the ‘1:2:1’ integrable case of the two-particle quartic potential. A general approach for finding elliptic solutions is given. New solutions which are associated with two-gap Treibich-Verdier potentials are found. General quasi-periodic solutions are given in terms of two dimensional theta functions with explicit expressions for frequencies in terms of theta constants. The reduction of quasi-periodic solutions to elliptic functions is discussed.

Uniform global asymptotic stability for half-linear differential systems with time-varying coefficients

2011 ◽  
Vol 141 (5) ◽  
pp. 1083-1101 ◽  
Author(s):  
Masakazu Onitsuka ◽  
Jitsuro Sugie
Keyword(s):  
Linear System ◽  
Zero Solution ◽  
Integrable Case ◽  
Time Varying ◽  
Globally Asymptotically Stable ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential ◽  
Image Position

The present paper deals with the following system:where p and p* are positive numbers satisfying 1/p + 1/p* = 1, and ϕq(z) = |z|q−2z for q = p or q = p*. This system is referred to as a half-linear system. We herein establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) for the zero solution to be uniformly globally asymptotically stable. If (e(t), f(t)) ≡ (h(t), g(t)), then the half-linear system is integrable. We consider two cases: the integrable case (e(t), f(t)) ≡ (h(t), g(t)) and the non-integrable case (e(t), f(t)) ≢ (h(t), g(t)). Finally, some simple examples are presented to illustrate our results.

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