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2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Vladimir Vyacheslavovich Sokolov
Keyword(s):  


2018 ◽  
Vol 23 (6) ◽  
pp. 785-796 ◽  
Author(s):  
Andrey V. Tsiganov
Keyword(s):  


2018 ◽  
Vol 190 ◽  
pp. 156-168
Author(s):  
Alexandru Buium ◽  
Emma Previato
Keyword(s):  


2017 ◽  
Vol 50 (24) ◽  
pp. 245203 ◽  
Author(s):  
Kinji Kimura
Keyword(s):  
Lax Pair ◽  


2017 ◽  
Vol 173 ◽  
pp. 37-63 ◽  
Author(s):  
Alexandru Buium ◽  
Emma Previato
Keyword(s):  


2016 ◽  
Vol 13 (04) ◽  
pp. 1650042 ◽  
Author(s):  
Sumanto Chanda ◽  
Partha Guha ◽  
Raju Roychowdhury

Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux–Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux–Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang–Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux–Halphen system and occurring in the study of Bianchi-IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.



2015 ◽  
Vol 27 (04) ◽  
pp. 1550011 ◽  
Author(s):  
Partha Guha

Recently, Kupershmidt [38] presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by Karasu-Kalkani et al. [31]. In this paper, we demonstrate that Kupershmidt's method can be interpreted as an infinite-dimensional analogue of the Euler–Poincaré–Suslov (EPS) formulation. In a finite-dimensional case, we modify Kupershmidt's deformation of the Euler top equation to obtain the standard EPS construction on SO(3). We extend Kupershmidt's infinite-dimensional construction to construct a nonholonomic deformation of a wide class of coupled KdV equations, where all these equations follow from the Euler–Poincaré–Suslov flows of the right invariant L2 metric on the semidirect product group [Formula: see text], where Diff (S1) is the group of orientation preserving diffeomorphisms on a circle. We generalize our construction to the two-component Camassa–Holm equation. We also give a derivation of a nonholonomic deformation of the N = 1 supersymmetric KdV equation, dubbed as sKdV6 equation and this method can be interpreted as an infinite-dimensional supersymmetric analogue of the Euler–Poincaré–Suslov (EPS) method.



Author(s):  
Richard H. Cushman ◽  
Larry M. Bates
Keyword(s):  


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