Hierarchy of WDVV associativity equations for n = 3 case and N = 2 when V0 6= 0

Abstract

We investigate solutions of Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV) equations. The article discusses nonlinear equations of the third order for a function f = f(x, t)) of two independent variables x, t . The solution of some cases of hierarchy equations of associativity illustrated. Lax pairs for the system of three equations, that contains the equation of associativity are written to find the hierarchy of associativity equation. Using the compatibility condition are
found the relations between the matrices U, V2, V1, V0 . The elements of matrix V2 are found with the expression of zij and independent and dependent variables for the matrix V2. Also solving elements of matrix V1 expressed through yij and independent and dependent variables for the
matrix V1 . The elements of matrix V0 are found with the expression of rij and independent and dependent variables for the matrix V0 too. Expressed are variables at , bt
, ct of three equations are written with the help of matrix elements zij , yij .