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Rational solutions for algebraic solitons in the massive Thirring model

Zhen Zhao, Cheng He, Baofeng Feng, Dmitry E. Pelinovsky

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 $\mathcal{O}(\sqrt{t})$.

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 . 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 -th member of the hierarchy describes a nonlinear superposition of algebraic solitons with identical masses and corresponds to an embedded eigenvalue of algebraic multiplicity . 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 with arbitrary parameters, which admits poles in the upper half-plane and poles in the lower half-plane. Assuming that the leading-order polynomials have exactly real roots, we show that the -th member of the hierarchy describes the slow scattering of algebraic solitons on the time scale .

Paper Structure

This paper contains 18 sections, 16 theorems, 242 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main results
  4. Comparison with previous works
  5. Methodology and organization of the paper
  6. Proof of Theorem \ref{['theorem-Wronskian']}
  7. Examples of solitons via double-Wronskian solutions
  8. Relation to double-Wronskian solutions of the two-component KP hierarchy
  9. Proof of Theorem \ref{['theorem-rat-sol']}
  10. Proof of Theorem \ref{['theorem-polyn']}
  11. Examples of the rational solutions
  12. Double algebraic soliton
  13. Triple algebraic soliton
  14. Quadruple algebraic soliton
  15. Quintuple and sextuple algebraic solitons
  16. ...and 3 more sections

Key Result

Theorem 1

Fix $N \in \mathbb{N}$. Let $A \in \mathbb{C}^{2N \times 2N}$ be an invertible matrix which can be factorized by $S \in \mathbb{C}^{2N \times 2N}$ in the form Define two vectors $\phi, \psi \in \mathbb{C}^{2N}$ from solutions of the linear equations subject to the relation Then, the following double-Wronskian functions satisfy the bilinear equations (MTM-4) with $C = (-{\rm i})^N/|S|$.

Figures (5)

  • Figure 1: Double algebraic solitons given by (\ref{['rat-2']}): the solution surface for $|u(x,t)|^2 + |v(x,t)|^2$ (left) and the density plot (right) together with the roots of the principal polynomial $p_2(x,t)$ (red curves).
  • Figure 2: Triple algebraic solitons given by (\ref{['rat-3']}): the solution surface for $|u(x,t)|^2 + |v(x,t)|^2$ (left) and the density plot (right) together with the roots of the principal polynomial $p_3(x,t)$ (red curves).
  • Figure 3: Quadruple algebraic solitons given by (\ref{['rat-4']}): the solution surface for $|u(x,t)|^2 + |v(x,t)|^2$ (left) and the density plot (right) together with the roots of the principal polynomial $p_4(x,t)$ (red curves).
  • Figure 4: Quintuple algebraic solitons: the solution surface for $|u(x,t)|^2 + |v(x,t)|^2$ (left) and the density plot (right) together with the roots of the principal polynomial $p_5(x,t)$ (red curves).
  • Figure 5: Sextuple algebraic solitons: the solution surface for $|u(x,t)|^2 + |v(x,t)|^2$ (left) and the density plot (right) together with the roots of the principal polynomial $p_6(x,t)$ (red curves).

Theorems & Definitions (39)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • Theorem 3
  • Remark 5
  • Remark 6
  • Lemma 1
  • ...and 29 more