Singular Miura type initial profiles for the KdV equation

TL;DR

This paper demonstrates that the KdV equation transforms certain singular initial profiles into meromorphic functions without real poles, revealing new insights into the evolution of singular solutions.

Contribution

It introduces a class of singular initial profiles for the KdV equation and shows their evolution into meromorphic functions with no real poles.

Findings

Singular initial profiles of the form q=r'+r^2 evolve into meromorphic functions.

The evolution preserves the absence of real poles in the solution.

The results extend understanding of the KdV flow for singular initial data.

Abstract

We show that the KdV flow evolves any real singular initial profile q of the form q=r'+r^2, where r\inL_{loc}^2, r|_{R_+}=0 into a meromorphic function with no real poles.