Multi-soliton solutions to the Thirring model through the reduction method
type: Item
instance of: scholarly article
title: Multi-soliton solutions to the Thirring model through the reduction method
zbMATH Open document ID: 0539.35071
DOI: 10.1007/BF00420038
author: N. E. Zubov
published in: Letters in Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: We show how to use the reduction method of A. V. Mikhajlov on a \(2\times 2\) Zakharov-Shabat system in order to get two types of Thirring models (which differ by demanding a hermitian or anti-hermitian prescription) up to gauge classes. Explicitly, these are described by \[ (1)\quad i\phi_{1,\eta}-2\phi_ 2-(\epsilon /2)| \phi_ 2|^ 2\phi_ 1=0,\quad i\phi_{2,\xi}-2\phi_ 1-(\epsilon /2)| \phi_ 1|^ 2\phi_ 2=0 \] where \(\epsilon =1,-1\), and they are the integrability conditions of a Zakharov-Shabat system of the form \[ (2) \quad \psi_{,\xi}=U\psi,\quad \psi_{,\eta}=V\psi,\quad U=U_ 0+U_ 1\lambda +U_ 2\lambda^ 2,\quad V=V_ 0+V_ 1\lambda^{-1}+V_ 2\lambda^{-2},\quad \lambda \in {\mathbb{C}}P^ 1,\quad U_ i,V_ i\in s1(2,{\mathbb{C}}). \] We then look for a gauge transformation \((\psi,U,V)\to({\hat \psi},\hat U,\hat V)\) and ask whether it be a Bäcklund mapping with respect to (1,2). This step is in fact the dressing of the vacuum solution for (1) and it involves a dressing matrix \(\chi(\xi,\eta,\lambda)\) which is meromorphic in the \(\lambda\) variable. By choosing an appropriate singular structure for \(\chi\), we are able to construct explicitly the whole solitonic class of solutions for (1).
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 35G20
zbMATH DE Number: 3857616
zbMATH Keywords: Dressing method
zbMATH Keywords: multisolitons
zbMATH Keywords: reduction method
zbMATH Keywords: Zakharov-Shabat system
zbMATH Keywords: Thirring models
zbMATH Keywords: Bäcklund mapping
MaRDI profile type: Publication
cites work: The soliton correlation matrix and the reduction problem for integrable systems
cites work: Quadratic pseudopotentials for Gl(N,\({\mathbb{C}})\) principal sigma models
full work available at URL: https://doi.org/10.1007/bf00420038
OpenAlex ID: W2001527147
description: scientific article; zbMATH DE number 3857616