Abstract
Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented by Kupershmidt [Phys. Lett. A 372, 2634 (2008)], 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: Korteweg–de Vries (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.
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