New computational formulae concerning the constant in the trace identity and the quadratic-form identity

The trace identity and the quadratic-form identity are all simple and powerful tools for establishing Hamiltonian structure of integrable hierarchies of soliton equations, the constant γ contained in the two identities are all to be determined. It has been a left problem to seek for computing formulas on γ , which had been specially proposed by Tu [Tu Guizhang. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30(2):330–8]. In this paper, we create an efficient method for obtaining γ by making use of two procedures. First, a few quadratic expressions G ( V ) ’s are discovered from the solvable conditions on Λ , where Λ satisfies the equation [ Λ , V ] - V λ = γ λ V , whereas, G ( V ) and γ have the clear relations. Second, by means of V x = [ U , V ] , we prove that G ( V ) is an one-place function with aspect to λ , but not related to x. It follows from the above two steps that the formula γ = - λ 2 d d λ ln | G ( V ) | is obtained. This technique is verified to be feasible and efficient by applying it to a few examples.