N ov 2 00 1 Singularity analysis of a new discrete nonlinear Schrödinger equation

We apply the Painlevé test for integrability to the new discrete nonlinear Schrödinger equation introduced by Leon and Manna. Since the singular expansions of solutions of this equation turn out to contain non-dominant logarithmic terms, we conclude that the studied equation is nonintegrable. This result supports the recent observation of Levi and Yamilov that the Leon–Manna equation does not admit high-order generalized symmetries. Recently, Leon and Manna [1] introduced the following new discrete non-linear Schrödinger equation: αψ m,tt = iβ (ψ m+1 − ψ m−1) + 2 |ψ m | 2 ψ m (1) where α and β are nonzero real parameters, m = 0, ±1, ±2,. .. . In [1], the equation (1) was derived from the integrable Toda lattice equation through the reductive perturbation analysis. More recently, Levi and Yamilov [2] studied the integrability of (1) by means of the generalized symmetry analysis , and found that the equation (1) does not admit local generalized symmetries of order higher than three and, hence, does not possess the same integrability properties as the Toda lattice equation, from which it has been derived. In the present short note, we study the integrability of the Leon–Manna equation (1) by means of the Painlevé test, following the Ablowitz–Ramani– Segur algorithm of singularity analysis [3] (see also [4]). We find the presence of non-dominant logarithmic terms in the singular expansions of solutions 1