Tri-Hamiltonian formulation for certain integrable lattice equations

Abstract

A systematic investigation of integrable differential–difference equations with two independent variables admitting multi-Hamiltonian structure is presented. Considering the Volterra (VL), Toda (TL), Relativistic Toda (RT), Belov–Chaltikian (BC) and Blaszak–Marciniak both three (BM3) and four (BM4) coupled lattice equations it is shown that they admit a sequence of operators out of which only three are Hamiltonian ones and so they are tri-Hamiltonian systems only. It is observed that the constructed third operator for VL and BC lattice equations is Hamiltonian only if the field variable is periodic with even period.