Recurrent procedure for the determination of the Free Energy $\epsilon^{2}$-expansion in the Topological String Theory

Abstract: We present here the iteration procedure for the determination of free energy

e p s i l o n^{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

e p s i l o n^{2}

-expansion of

u (x, t_{1}, t_{2}, . . .)

after the infinite number of shifts of

u (x, 0, 0, . . .) e q u i v 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)

.

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