Direct Scattering for the KdV Equation with a Step-like Finite-Gap Potential: A Riemann--Hilbert Approach

Abstract

We develop the direct scattering theory for the KdV equation with step-like finite-gap backgrounds under perturbations. More precisely, we consider initial data that asymptotically approach two distinct one-gap periodic travelling wave solutions as \(x \to \pm \infty\). Under suitable assumptions on the perturbation, we formulate the direct scattering problem and establish the analytic structure of the associated scattering data. In particular, we reformulate the problem in terms of a vector Riemann--Hilbert problem, which provides a foundation for the study of long-time asymptotics of perturbed finite-gap potentials. This formulation highlights the connection between step-like finite-gap scattering theory and the Riemann--Hilbert framework arising in soliton-gas type settings.