Non-linear WKB Analysis of the String Equation

TL;DR

This paper uses non-linear WKB analysis to approximate solutions of the string equation, demonstrating excellent agreement with numerical results and revealing physical solutions as separatrices on degenerate Riemann surfaces.

Contribution

It introduces a non-linear WKB method for the string equation and shows its high accuracy compared to numerical solutions, providing new analytical insights.

Findings

Non-linear WKB solutions closely match numerical results.

Physical solutions correspond to separatrices on degenerate Riemann surfaces.

Analytic approximation agrees well with previous numerical solutions for specific cases.

Abstract

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.