We apply non-linear WKB analysis to the study of the string equation. Even though the solutions obtained with this method are not exact, they approximate extremely well the true solutions, as we explicitly show using numerical simulations. "Physical" solutions are seen to be separatrices corresponding to degenerate Riemann surfaces. We obtain an analytic approximation in excellent agreement with the numerical solution found by Parisi et al. for the k = 3 case.
Fucito, F., Gamba, A., Martellini, M., Ragnisco, O. (1992). NONLINEAR WKB ANALYSIS OF THE STRING EQUATION. INTERNATIONAL JOURNAL OF MODERN PHYSICS B, 6(11-12), 2123-2147 [10.1142/S0217979292001067].