Exponential and algebraic double-soliton solutions of the massive Thirring model

TL;DR

This paper classifies and constructs exponential and algebraic double-soliton solutions of the massive Thirring model, clarifying their spectral origins and resolving a long-standing conjecture about embedded eigenvalues.

Contribution

It introduces explicit constructions of double-soliton solutions linked to different spectral eigenvalues, advancing understanding of the inverse scattering transform for the model.

Findings

Exponential double-solitons correspond to double isolated eigenvalues.

Algebraic double-solitons relate to double embedded eigenvalues.

Resolved the conjecture on the existence of multiple embedded eigenvalues.

Abstract

The newly discovered exponential and algebraic double-soliton solutions of the massive Thirring model in laboratory coordinates are placed in the context of the inverse scattering transform. We show that the exponential double-solitons correspond to double isolated eigenvalues in the Lax spectrum, whereas the algebraic double-solitons correspond to double embedded eigenvalues on the imaginary axis, where the continuous spectrum resides. This resolves the long-standing conjecture that multiple embedded eigenvalues may exist in the spectral problem associated with the massive Thirring model. To obtain the exponential double-solitons, we solve the Riemann--Hilbert problem with the reflectionless potential in the case of a quadruplet of double poles in each quadrant of the complex plane. To obtain the algebraic double-solitons, we consider the singular limit where the quadruplet of double poles degenerates into a symmetric pair of double embedded poles on the imaginary axis.