Lie Algebra, Bi-Hamiltonian Structure and Reduction Problem for Integrable Nonlinear Systems

Abstract

Bi-Hamiltonian structures of integrable nonlinear evolution equations are reviewed in the framework of infinite dimensional Lie Algebra. We have shown that it is actually possible to derive the two-symplectic structures from the formalism of G. Zhang Tu et al. Furthermore we have extended their formalism to system with 3 × 3 matrix structure. An important aspect of our formulation is to implement reduction mechanism to arrive at a specific nonlinear system. As examples we have discussed the cases of KdV, Langmuir solitons and generalised Lund-Regge system, and Heisenberg Spin Chain. The method is explained on the basis of the above examples and it is shown that it is possible to reproduce all the previous results by this Lie algebraic technique.