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

Abstract

Dedicated to Professor Victor Kac on his 70th birthday with great respect and admirationRecently, Kupershmidt [38] presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by . 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 L 2 metric on the semidirect product group Diff(S 1 ) C ∞ (S 1 ), where Diff(S 1 ) 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.

A Study of Nonholonomic Deformations of Nonlocal Integrable Systems Belonging to the Nonlinear Schrödinger Family

“…Such an equation is subjected to the dynamics of equations (30) and (31), and therefore do not yield any new dynamics, and eventually reflects the constraint itself in a different form. This is in accord with the previous argument that no term, with power of λ other than that responsible for yielding equations (30) and (31), can yield dynamics to the non-local NLS system, as it will violate the overall integrability of the system itself.…”

“…The non-holonomic deformation of derivative NLS and Lenells-Fokas equations was discussed in [29], while such deformation of generalized KdV type equations was taken up in [30] where emphasis was put on the geometrical aspect of the problem. Kupershmidt's infinite-dimensional construction was extended in [31] to obtain non-holonomic deformation of a wide class of coupled KdV systems, all of which are generated from the Euler-Poincare-Suslov flows. A comparative study encompassing two different types of deformations, viz.…”

Two‐level linearized and local uncoupled difference schemes for the two‐component evolutionary Korteweg‐de Vries system

“…In , two‐component evolutionary systems of homogeneous KdV equations of second and third order were studied by applying homotopy analysis method (HAM). In , a nonholonomic deformation of a wide class of coupled KdV equations was constructed by extending Kupershmidt's infinite‐dimensional construction and these equations were analyzed by Euler‐Poincaré‐Suslov method. In , two‐component KdV system was studied by prolongation technique and Painlevé analysis.…”

Study of quasi-integrable and non-holonomic deformation of equations in the NLS and DNLS hierarchy

“…NHD for the coupled KdV system was thereby generated. In [28] the author extended Kupershmidt's infinite-dimensional construction to generate NHD of a wide class of coupled KdV systems, all of which follow from the Euler-Poincaré-Suslov flows.…”

“…Conservation properties of these QI systems are demonstrated mostly via numerical methods [14,15,16,17,18,19,29] for the lower order hierarchical equations. On the other hand nonholonomically deformed systems remain completely integrable [23,24,25,26,27,28]. The fact that the deformation is applied to the temporal Lax component automatically preserves the scattering data of the undeformed system [24,25,26].…”