The inverse scattering transform for the KdV equation with step-like singular Miura initial profiles

TL;DR

This paper develops an inverse scattering transform for the KdV equation with singular initial data, showing that solutions are meromorphic with no real poles for positive times, advancing understanding of integrable systems with singularities.

Contribution

It introduces a method to handle singular initial profiles in the KdV equation using inverse scattering, extending classical results to more general initial data.

Findings

Solution $q(x,t)$ is meromorphic for all $t>0$

No real poles in the solution for positive times

Extends inverse scattering methods to singular initial data

Abstract

We develop the inverse scattering transform for the KdV equation with real singular initial data $q(x)$ of the form $q(x) = r'(x) + r(x)^2$, where $r\in L^2_{\textrm{loc}}$ and $r=0$ on $\mathbb R_+$. As a consequence we show that the solution $q(x,t)$ is a meromorphic function with no real poles for any $t>0$.