Rational solutions for algebraic solitons in the massive Thirring model

TL;DR

This paper studies the hierarchy of rational solutions in the massive Thirring model, revealing their polynomial structure, pole distribution, and describing slow soliton scattering phenomena.

Contribution

It provides a rigorous proof that rational solutions are characterized by degree N^2 polynomials with specific pole configurations, advancing understanding of algebraic solitons in the MTM.

Findings

Solutions are defined by degree N^2 polynomials with 2N and N(N±1)/2 poles in upper and lower half-planes.

Each solution describes N algebraic solitons with identical masses.

The N-th solution models slow scattering of N solitons on a √t time scale.

Abstract

An algebraic soliton of the massive Thirring model (MTM) is expressed by the simplest rational solution of the MTM with the spatial decay of $\mathcal{O}(x^{-1})$. The corresponding potential is related to a simple embedded eigenvalue in the Kaup--Newell spectral problem. This work focuses on the hierarchy of rational solutions of the MTM, in which the $N$-th member of the hierarchy describes a nonlinear superposition of $N$ algebraic solitons with identical masses and corresponds to an embedded eigenvalue of algebraic multiplicity $N$. We show that the hierarchy of rational solutions can be constructed by using the double-Wronskian determinants. The novelty of this work is a rigorous proof that each solution is defined by a polynomial of degree $N^2$ with $2N$ arbitrary parameters, which admits $\frac{N (N-1)}{2}$ poles in the upper half-plane and $\frac{N(N+1)}{2}$ poles in the lower half-plane. Assuming that the leading-order polynomials have exactly $N$ real roots, we show that the $N$-th member of the hierarchy describes the slow scattering of $N$ algebraic solitons on the time scale $\mathcal{O}(\sqrt{t})$.