A differential-difference system related to toroidal Lie algebra (Q2773088)

This paper is devoted to the following differential-difference system of the form NEWLINE\[NEWLINE\begin{cases} \partial_tu_k=\Delta_{-k}\left( {\partial_x u_{k+1}\over 1-\exp(-u_{k+1}-u_k)} -{\partial_xu_k\over 1-\exp (u_{k+1} +u_k)}-{1+\exp(u_{k+1}+u_k) \over 1-\exp(u_{k+1} +u_k)}v_k \right)\\ \Delta_{-k}v_k={\partial_xu_{k+1}\over u_{k+1}}+{\partial_x(u_{k+1}+u_k)\over 1-\exp(u_{k+1}+u_k)}+{\partial_xu_k\over u_k}+{\partial_x(u_k+u_{k-1})\over 1-\exp(u_k+u_{k-1}} \end{cases} \tag{1}NEWLINE\]NEWLINE where \(\Delta_{-k}\) denotes the backward-difference operator, that is \(\Delta_{-k}= u_k-u_{k-1}\). The authors show that (1) has soliton-type solutions. The symmetry of the system is given by the toroidal Lie algebra \(sl^{\text{tor}}_2\).