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Existence of global solutions to the massive Thirring model in the non-laboratory coordinates

Sucai Niu, Junyi Zhu, Xueru Wang

TL;DR

This work extends the inverse scattering transform to the massive Thirring model in non-laboratory coordinates, constructing a two‑transformation framework that yields two Jost function systems and associated Riemann-Hilbert problems. By proving solvability and Lipschitz continuity at both the direct and inverse stages, the authors obtain explicit reconstructions for $v$ and, via time evolution and the dressing method, for $u$, along with conservation laws. Under smallness and spectral-constraint assumptions (no eigenvalues or resonances), they establish unique global solutions with Lipschitz dependence on the initial scattering data, and show that the relevant norms of $v$ and $u$ remain controlled globally in time. The results provide a rigorous global well-posedness theory for the MT model in non-laboratory coordinates and connect spectral data evolution to explicit potential reconstructions, with implications for soliton dynamics and conserved quantities in this integrable system.

Abstract

The massive Thirring model in the non-laboratory coordinates is considered by the Riemann-Hilbert approach. Existence of global solutions is shown for the cases of the associated Riemann-Hilbert problem without eigenvalues or resonances. The Lipschitz continuity of the map from the potential $v_0(x)\in H^2(\mathbb{R})\cap H^{1,1}(\mathbb{R})$ to the scattering data is given in the direct scattering transform. Two transform matrices are introduced to curb the convergence of the Volterra integral equations and the relevant estimates of the modified Jost functions. For small potential, the solvability of the Riemann-Hilbert problems without eigenvalues or resonances is discussed. The Lipschitz continuity of the map from the scattering data to the potential $v(x)$ is shown. The reconstructions for potential $u(x,t)$ and $v(x,t)$ are finished by considering the time dependence of the scattering data and by constructing the conservation laws obtain via the dressing method.

Existence of global solutions to the massive Thirring model in the non-laboratory coordinates

TL;DR

This work extends the inverse scattering transform to the massive Thirring model in non-laboratory coordinates, constructing a two‑transformation framework that yields two Jost function systems and associated Riemann-Hilbert problems. By proving solvability and Lipschitz continuity at both the direct and inverse stages, the authors obtain explicit reconstructions for and, via time evolution and the dressing method, for , along with conservation laws. Under smallness and spectral-constraint assumptions (no eigenvalues or resonances), they establish unique global solutions with Lipschitz dependence on the initial scattering data, and show that the relevant norms of and remain controlled globally in time. The results provide a rigorous global well-posedness theory for the MT model in non-laboratory coordinates and connect spectral data evolution to explicit potential reconstructions, with implications for soliton dynamics and conserved quantities in this integrable system.

Abstract

The massive Thirring model in the non-laboratory coordinates is considered by the Riemann-Hilbert approach. Existence of global solutions is shown for the cases of the associated Riemann-Hilbert problem without eigenvalues or resonances. The Lipschitz continuity of the map from the potential to the scattering data is given in the direct scattering transform. Two transform matrices are introduced to curb the convergence of the Volterra integral equations and the relevant estimates of the modified Jost functions. For small potential, the solvability of the Riemann-Hilbert problems without eigenvalues or resonances is discussed. The Lipschitz continuity of the map from the scattering data to the potential is shown. The reconstructions for potential and are finished by considering the time dependence of the scattering data and by constructing the conservation laws obtain via the dressing method.

Paper Structure

This paper contains 10 sections, 31 theorems, 365 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The direct scattering transform and Lipschitz continuity
  3. Riemann-Hilbert problems
  4. The inverse scattering transform and Lipschitz continuity
  5. Solvability of the Riemann-Hilbert problem
  6. Reconstruction for the potential
  7. Conservation laws and reconstruction for potential
  8. Dressing method and conservation laws
  9. Reconstruction for the potential
  10. Global solutions to the MT model

Key Result

Lemma 1

Let $v\in L^2(\mathbb{R})\cap L^1(\mathbb{R}), v_x\in L^1(\mathbb{R})$, and the condition mtb8b is valid. For every $z\in\mathbb{R}$, there exist unique solutions $m_\pm(\cdot;z)\in L^\infty(\mathbb{R}), n_\pm(\cdot;z)\in L^\infty(\mathbb{R})$ to the integral equations mtb6 and mtb7. For every $x\in

Figures (2)

  • Figure 1: If $f(k)$ is an even function, the orientation of the integral contour is defined in left; if $f(k)$ is an odd function, the orientation of the integral contour is defined in right.
  • Figure 2: The scheme for the proof of Theorem \ref{['the1']}.

Theorems & Definitions (31)

  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Proposition 2
  • ...and 21 more