Nonholonomic deformation of coupled and supersymmetric KdV equations and Euler-Poincaré-Suslov method

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Abstract

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 {\widehat {{\rm Diff}(S1) \ltimes C^{\infty}(S1)}}, 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.

Original languageBritish English
Article number1550011
JournalReviews in Mathematical Physics
Volume27
Issue number4
DOIs
StatePublished - 29 May 2015

Keywords

  • Bott-Virasoro group
  • coupled KdV equations
  • Diffeomorphism
  • geodesic flows
  • Kuper-KdV equations
  • nonholonomic deformation
  • superconformal group

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