Inverse Scattering Transform for the Massive Thirring Model: Delving into Higher-Order Pole Dynamics

TL;DR

This paper develops a novel inverse scattering transform method for the massive Thirring model, addressing higher-order pole cases and establishing solution existence, uniqueness, and reconstruction techniques.

Contribution

It introduces a new approach to handle higher-order poles in the inverse scattering transform for the massive Thirring model, including spectral problem analysis and Riemann--Hilbert problem formulation.

Findings

Constructed a Riemann--Hilbert problem with residue conditions for higher-order poles.

Proved existence and uniqueness of the solution to the pole-free Riemann--Hilbert problem.

Provided a method to reconstruct solutions in the reflectionless case via linear algebraic systems.

Abstract

We investigate the inverse scattering problem for the massive Thirring model, focusing particularly on cases where the transmission coefficient exhibits $N$ pairs of higher-order poles. Our methodology involves transforming initial data into scattering data via the direct scattering problem. Utilizing two parameter transformations, we examine the asymptotic properties of the Jost functions at both vanishing and infinite parameters, yielding two equivalent spectral problems. We subsequently devise a mapping that translates the obtained scattering data into a $2 \times 2$ matrix Riemann--Hilbert problem, incorporating several residue conditions at $N$ pairs of multiple poles. Additionally, we construct an equivalent pole-free Riemann--Hilbert problem and demonstrate the existence and uniqueness of its solution. In the reflectionless case, the $N$-multipole solutions can be reconstructed by resolving two linear algebraic systems.