New computational formulae concerning the constant γγ 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 γγ contained in the two identities are all to be determined. It has been a left problem to seek for computing formulas on γγ, 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 γγ by making use of two procedures. First, a few quadratic expressions G(V)G(V)’s are discovered from the solvable conditions on ΛΛ, where ΛΛ satisfies the equation [Λ,V]-Vλ=γλV, whereas, G(V)G(V) and γγ have the clear relations. Second, by means of Vx=[U,V]Vx=[U,V], we prove that G(V)G(V) is an one-place function with aspect to λλ, but not related to x  . It follows from the above two steps that the formula γ=-λ2ddλln|G(V)| is obtained. This technique is verified to be feasible and efficient by applying it to a few examples.