0 M ar 1 99 9 Recurrent procedure for the determination of the Free Energy ǫ 2 - expansion in the Topological String

We present here the iteration procedure for the determination of free energy ǫ 2-expansion using the theory of KdV-type equations. In our approach we use the conservation laws for KdV-type equations depending explicitly on times t 1 , t 2 ,. .. to find the ǫ 2-expansion of u(x, t 1 , t 2 ,. . .) after the infinite number of shifts of u(x, 0, 0,. . .) ≡ x along t 1 , t 2 ,. .. in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to u n (x). This work is devoted mainly to the calculation of low-dispersion expansions of the solutions of KdV type equations and their using for calculation of Weil-Petersson volumes of moduli spaces. More precisely we obtain a recurrent procedure for the formulas presented in [2] for such expansions and will refer here to the papers [2] and [3] where the more detailed information and references can be found. Our procedure is based on the quasi-classical expansion for Schrödinger operator and uses also the times-dependent inte-grals for KdV type equations which, as far as we know, were not mentioned in the previous papers.