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

TL;DR

This paper proves the existence of global solutions to the massive Thirring model in non-laboratory coordinates using Riemann-Hilbert methods, focusing on cases without eigenvalues or resonances and analyzing the scattering transform.

Contribution

It introduces a Riemann-Hilbert approach to establish global solutions and analyzes the Lipschitz continuity of the scattering transform for the massive Thirring model.

Findings

Global solutions exist without eigenvalues or resonances.

Lipschitz continuity of the scattering map is established.

Constructed conservation laws via the dressing method.

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.