Nonholonomic deformation of generalized KdV-type equations

Abstract

Karasu-Kalkani et al (2008 J. Math. Phys. 49 073516) recently derived a new sixth-order wave equation KdV6, which was shown by Kupershmidt (2008 Phys. Lett. 372A 2634) to have an infinite commuting hierarchy with a common infinite set of conserved densities. Incidentally, this equation was written for the first time by Calogero and is included in the book by Calogero and Degasperis (1982 Lecture Notes in Computer Science vol 144 (Amsterdam: North-Holland) p 516). In this paper, we give a geometric insight into the KdV6 equation. Using Kirillov's theory of coadjoint representation of the Virasoro algebra, we show how to obtain a large class of KdV6-type equations equivalent to the original equation. Using a semidirect product extension of the Virasoro algebra, , we propose the nonholonomic deformation of the Ito equation. We also show that the Adler–Kostant–Symes scheme provides a geometrical method for constructing nonholonomic deformed integrable systems. Applying the Adler–Kostant–Symes scheme to loop algebra, we construct a new nonholonomic deformation of the coupled KdV equation.

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

“…In a recent paper, Yao and Zeng [57] showed that the KdV6 equation is equivalent to the Rosochatius deformation of KdV equation with self-consistent sources. In our earlier paper [21], we extended Yao and Zeng's result to construct many other equations equivalent to the KdV6 equation and we identified that the constraint equation of w is a stabilizer equation of the Virasoro orbit. We tacitly replaced this equation with an equivalent partner equation to obtain various new avatars of the KdV6 equation.…”

“…We put it in a more systematic form using Kirillov's coadjoint orbit method. In [21], we extended Kuperschmidt's formalism [38] to extended Virasoro algebra Vir C ∞ (S 1 ) to construct the Ito6 equation. It a Thanks to Professor Francesco Calogero for sharing this information.…”

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 constraint of equation (35) is non-holonomic in nature, as it contains differentials of corresponding variables, and characterizes the corresponding deformation.…”

“…In [28] the work was extended to include the non-holonomic deformation of both KdV and mKdV equations along with their symmetries, hierarchies and integrability. 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.…”

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

“…The work was extended in [25] to include the NHD of both KdV and mKdV equations as well as their symmetries, hierarchies and integrability. While studies on the non-holonomic deformation of DNLS and Lenells-Fokas equations were carried out in [26], NHD of generalized KdV type equations was discussed in [27], wherein a geometric angle was provided into the KdV6 equation. In this work, Kirrilov's theory of co-adjoint representation of the Virasoro algebra was used to generate a large class of KdV6 type equations equivalent to the original equation.…”