On the inverse scattering transform for the KdV equation with summable initial data

TL;DR

This paper develops a rigorous inverse scattering method for solving the KdV equation with a class of initial data that are summable and supported on the positive real line, extending previous short-range results.

Contribution

It introduces a new inverse scattering framework for the KdV equation with summable, half-line supported initial data using Hankel operators and reflection coefficients.

Findings

Derived a trace-type representation for solutions of the KdV equation.

Established convergence and continuity properties of Hankel operators for the problem.

Extended inverse scattering techniques beyond the standard short-range setting.

Abstract

We consider the Cauchy problem for the Korteweg--de Vries equation with real initial data $q$ that is both $L^1$ and $L^2$ summable and supported on (0,\infty). Using the left reflection coefficient and Hankel operators on the Hardy space $H^2$, we derive a trace-type representation for the corresponding solution. The proof is based on approximation by compactly supported potentials, uniform convergence of the associated reflection coefficients away from the origin, and continuity properties of the resulting Hankel operators. This yields a rigorous inverse scattering construction for a class of summable half-line supported initial data beyond the standard short-range setting.