Multi-soliton solutions to the Thirring model through the reduction method
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).
scientific article; zbMATH DE number 3857616