TY - JOUR

T1 - Nonholonomic deformation of coupled and supersymmetric KdV equations and Euler-Poincaré-Suslov method

AU - Guha, Partha

N1 - Publisher Copyright:
© 2015 World Scientific Publishing Company.

PY - 2015/5/29

Y1 - 2015/5/29

N2 - 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.

AB - 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.

KW - Bott-Virasoro group

KW - coupled KdV equations

KW - Diffeomorphism

KW - geodesic flows

KW - Kuper-KdV equations

KW - nonholonomic deformation

KW - superconformal group

UR - http://www.scopus.com/inward/record.url?scp=84930069522&partnerID=8YFLogxK

U2 - 10.1142/S0129055X15500117

DO - 10.1142/S0129055X15500117

M3 - Article

AN - SCOPUS:84930069522

SN - 0129-055X

VL - 27

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

IS - 4

M1 - 1550011

ER -