Zhen Zhao, Cheng He, Baofeng Feng, and Dmitry E. Pelinovsky
Rational solutions for algebraic solitons in the massive Thirring model
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 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 N(N-1)/2 poles in the upper half-plane and 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 O(t^{1/2}).
Keywords:
massive Thirring model; algebraic solitons; Kaup-Newell spectral problem, embedded eigenvalues,
double-Wronskian solutions.