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

✍ Scribed by Fu-Kui Guo

• Book ID: 104012324
• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 209 KB
• Volume: 14
• Category: Article
• ISSN: 1007-5704
• DOI: 10.1016/j.cnsns.2008.03.011

✦ Synopsis

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 c contained in the two identities are all to be determined. It has been a left problem to seek for computing formulas on c, 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 c by making use of two procedures. First, a few quadratic expressions GðVÞ's are discovered from the solvable conditions on K, where K satisfies the equation

k V, whereas, GðVÞ and c have the clear relations. Second, by means of V x ¼ ½U; V, we prove that GðVÞ is an one-place function with aspect to k, but not related to x. It follows from the above two steps that the formula c ¼ À k 2 d dk ln jGðVÞj is obtained. This technique is verified to be feasible and efficient by applying it to a few examples.