Toda Lattice and Tomimatsu-Sato Solutions

We discuss an analytic proof of a conjecture (Nakamura) that solutions of Toda molecule equation give those of Ernst equation giving Tomimatsu-Sato solutions of Einstein equation. Using Pfaffian identities it is shown for Weyl solutions completely and for generic cases partially. April 1995 Rits-TH-9511 TOHO-FP-9551 Many of integrable non linear models are obtained from (anti) self-dual Yang-Mills equations by reduction [1](Ward's conjecture). Toda Lattice and Ernst equations are such examples of integrable systems. However the relationship between these integrable systems is not so simple. It is not affirmative that the system of Ernst equation belongs to the conventional category of integrable systems. For instance, Lax pair for the Ernst equation is not exactly the same type as those in typical soliton theories[2]. On the other hand, "direct method" developed by Hirota[3] has shown to be a very powerful tool in soliton theories. One of the advantages of the direct method amongst o...