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

TL;DR

This paper introduces a generalized Kupershmidt deformation method to generate new integrable systems from bi-Hamiltonian systems, extending existing techniques and verifying the conjecture that integrability is preserved in key examples.

Contribution

It proposes a new generalized deformation approach that preserves integrability and provides a systematic way to derive new integrable systems from known bi-Hamiltonian systems.

Findings

Verified the conjecture in KdV, Boussinesq, Jaulent-Miodek, and Camassa-Holm equations.

Presented a procedure to convert deformations into integrable Rosochatius systems.

Derived new integrable systems using the generalized Kupershmidt deformation.

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.