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arXiv · 1005.0281

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

Yuqin Yao·Yunbo Zeng
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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.

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Yuqin Yao, Yunbo Zeng. 2010-05-03. The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems. https://doi.org/10.1063/1.3431967

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