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

Abstract

Hamiltonian structures of integrable nonlinear evolution equations are studied 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. Furthermore we have extended this formalism to a 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.