Inverse scattering problem for the third-order equation on the line

TL;DR

This paper develops a method to solve the inverse scattering problem for a third-order linear differential equation with Schwartz class potentials, using Riemann-Hilbert problems and integral equations, to recover potentials from scattering data.

Contribution

It introduces a novel inverse scattering framework for third-order equations, including Riemann-Hilbert formulation and linear integral equations for potential recovery.

Findings

Explicit solutions for scattering coefficients and bound states.

A Riemann-Hilbert problem formulation for potential reconstruction.

A Marchenko-type integral equation for the case without bound states.

Abstract

We consider the third-order linear differential equation $$\displaystyle\frac{d^3\psi}{dx^3}+Q(x)\,\displaystyle\frac{d\psi}{dx}+P(x)\,\psi=k^3\,\psi,\qquad x\in\mathbb R,$$ where the complex-valued potentials $Q$ and $P$ are assumed to belong to the Schwartz class. We describe the basic solutions, the scattering coefficients, and the bound-state information, and we introduce the dependency constants and the normalization constants at the bound states. When the secondary reflection coefficients are zero, we provide a method to solve the corresponding inverse scattering problem, where the goal is to recover the two potentials $Q$ and $P$ from the scattering data set consisting of the transmission and primary reflection coefficients and the bound-state information. We formulate the corresponding inverse scattering problem as a Riemann--Hilbert problem on the complex $k$-plane and describe how the potentials are recovered from the solution to the Riemann--Hilbert problem. In the absence of bound states, we introduce a linear integral equation, which is the analog of the Marchenko integral equation used in the inverse scattering theory for the full-line Schr\"odinger equation. We describe the recovery of the two potentials from the solution to the aforementioned linear integral equation.