On a new integrable discretization of the derivative nonlinear Schr\"odinger (Chen-Lee-Liu) equation

Abstract

We propose a general integrable lattice system involving some free parameters, which contains known integrable lattice systems such as the Ablowitz-Ladik discretization of the nonlinear Schr\"odinger (NLS) equation as special cases. With a suitable choice of the parameters, it provides a new integrable space-discretization of the derivative NLS equation known as the Chen-Lee-Liu equation. Analogously to the continuous case, the space-discrete Chen-Lee-Liu system possesses a Lax pair and admits a complex conjugation reduction between the two dependent variables. Thus, we obtain a proper space-discretization of the Chen-Lee-Liu equation defined on the three lattice sites $n-1$, $n$, $n+1$ for the first time. Considering a negative flow of the discrete Chen-Lee-Liu hierarchy, we obtain a proper discretization of the massive Thirring model in light-cone coordinates. Multicomponent generalizations of the obtained discrete equations are straightforward because the performed computations are valid for the general case where the dependent variables are vector- or matrix-valued.