The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems

Abstract

Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented in [4], we propose the generalized Kupershmidt deformation to construct new systems from integrable bi-Hamiltonian systems, which provides a nonholonomic perturbation of the bi-Hamiltonian systems. The generalized Kupershmidt deformation is conjectured to preserve integrability. The conjecture is verified in a few representative cases: KdV equation, Boussinesq equation, Jaulent-Miodek equation and Camassa-Holm equation. For these specific cases, we present a general procedure to convert the generalized Kupershmidt deformation into the integrable Rosochatius deformation of soliton equation with self-consistent sources, then to transform it into a $t$-type bi-Hamiltonian system. By using this generalized Kupershmidt deformation some new integrable systems are derived. In fact, this generalized Kupershmidt deformation also provides a new method to construct the integrable Rosochatius deformation of soliton equation with self-consistent sources.

21 pages, to appear in Journal of Mathematical Physics

References in corpus (5)

• KdV6: An Integrable System
• A new integrable generalization of the Korteweg - de Vries equation
• The bi-Hamiltonian structure and new solutions of KdV6 equation
• Integrable Rosochatius deformations of higher-order constrained flows and the soliton hierarchy with self-consistent sources
• Integrability of Kupershmidt deformations

Cited by in corpus (3)

• Camassa-Holm and M-CIV equations with self-consistent sources: geometry and peakon solutions
• Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators
• The generalized Kupershmidt deformation for constructing new discrete integrable systems