scholarly journals Integrable Deformations of Foliations: a Generalization of Ilyashenko's Result

2021 ◽  
Vol 21 (2) ◽  
pp. 271-286
Author(s):  
Dominique Cerveau ◽  
Bruno Scárdua
Keyword(s):  
Integrable Deformations

Integrable deformations in the algebra of pseudodifferential operators from a Lie algebraic perspective

2013 ◽  
Vol 174 (1) ◽  
pp. 134-153 ◽  
Author(s):  
G. F. Helminck ◽  
A. G. Helminck ◽  
E. A. Panasenko
Keyword(s):  
Pseudodifferential Operators ◽  
Integrable Deformations

scholarly journals Integrable deformations of the XXZ spin chain

2013 ◽  
Vol 2013 (09) ◽  
pp. P09028 ◽  
Author(s):  
Niklas Beisert ◽  
Lucas Fiévet ◽  
Marius de Leeuw ◽  
Florian Loebbert
Keyword(s):  
Spin Chain ◽  
Integrable Deformations

scholarly journals A REVIEW OF INTEGRABLE DEFORMATIONS IN AdS/CFT

2007 ◽  
Vol 22 (13) ◽  
pp. 915-930 ◽  
Author(s):  
IAN SWANSON
Keyword(s):  
String Theory ◽  
Bethe Ansatz ◽  
Energy Spectra ◽  
Yang Mills ◽  
Twisted String ◽  
Pp Wave ◽  
Integrable Deformations ◽  
Type Iib String Theory ◽  
Bethe Equations ◽  
Wave Limit

Marginal β deformations of [Formula: see text] super-Yang–Mills theory are known to correspond to a certain class of deformations of the S5 background subspace of type IIB string theory in AdS5×S5. An analogous set of deformations of the AdS5 subspace is reviewed here. String energy spectra computed in the near-pp-wave limit of these backgrounds match predictions encoded by discrete, asymptotic Bethe equations, suggesting that the twisted string theory is classically integrable in this regime. These Bethe equations can be derived algorithmically by relying on the existence of Lax representations, and on the Riemann–Hilbert interpretation of the thermodynamic Bethe ansatz. This letter is a review of a seminar given at the Institute for Advanced Study, based on research completed in collaboration with McLoughlin.

scholarly journals Doubled aspects of generalised dualities and integrable deformations

2019 ◽  
Vol 2019 (2) ◽  
Author(s):  
Saskia Demulder ◽  
Falk Hassler ◽  
Daniel C. Thompson
Keyword(s):  
Integrable Deformations

Integrable deformations, bi-Hamiltonian structures and nonintegrability of a generalized Rikitake system

2019 ◽  
Vol 16 (04) ◽  
pp. 1950059 ◽  
Author(s):  
Kaiyin Huang ◽  
Shaoyun Shi ◽  
Zhiguo Xu
Keyword(s):  
Galois Groups ◽  
Point Of View ◽  
Integrable Case ◽  
Analytic First Integrals ◽  
Rikitake System ◽  
Constants Of Motion ◽  
Integrable Deformations ◽  
Particular Solution ◽  
Almost All ◽  
Parameter Values

The aim of this paper is to investigate a generalized Rikitake system from the integrability point of view. For the integrable case, we derive a family of integrable deformations of the generalized Rikitake system by altering its constants of motion, and give two classes of Hamilton–Poisson structures which implies these integrable deformations, including the generalized Rikitake system, are bi-Hamiltonian and have infinitely many Hamilton–Poisson realizations. By analyzing properties of the differential Galois groups of normal variational equations (NVEs) along certain particular solution, we show that the generalized Rikitake system is not rationally integrable in an extended Liouville sense for almost all parameter values, which is in accord with the fact that this system admits chaotic behaviors for a large range of its parameters. The non-existence of analytic first integrals are also discussed.

scholarly journals On the integrable deformations of a system related to the motion of two vortices in an ideal incompressible fluid

2019 ◽  
Vol 29 ◽  
pp. 01015 ◽  
Author(s):  
Cristian Lăzureanu ◽  
Ciprian Hedrea ◽  
Camelia Petrişor
Keyword(s):  
Incompressible Fluid ◽  
Mechanical System ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Mechanical Systems ◽  
Ideal Incompressible Fluid ◽  
First Integrals ◽  
Integrable Deformation ◽  
Integrable Deformations ◽  
The Given

Altering the first integrals of an integrable system integrable deformations of the given system are obtained. These integrable deformations are also integrable systems, and they generalize the initial system. In this paper we give a method to construct integrable deformations of maximally superintegrable Hamiltonian mechanical systems with two degrees of freedom. An integrable deformation of a maximally superintegrable Hamiltonian mechanical system preserves the number of first integrals, but is not a Hamiltonian mechanical system, generally. We construct integrable deformations of the maximally superintegrable Hamiltonian mechanical system that describes the motion of two vortices in an ideal incompressible fluid, and we show that some of these integrable deformations are Hamiltonian mechanical systems too.

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