## ] o c n m p [## Open Communications in Nonlinear Mathematical Physics |

Articles and Letters published during 2023 (not yet complete).

We present an application of the full-deautonomisation method to a class of secondorder mappings which, using an ancillary variable, can be cast into a form that greatly facilitates the study of their singularities. The ancillary approach was originally introduced to make it possible to construct discrete Painlevé equations associated with the affine Weyl group E (1) 8 by deautonomising a QRT mapping. The full-deautonomisation method has been shown to offer a practical technique for calculating the exact dynamical degree of a mapping, whereby allowing the detection of discrete integrability using only singularity analysis. We study the confinement property for a given singularity, for a wide class of mappings that includes the autonomous limit of the standard additive Painlevé equation with E (1) 8 symmetry. This leads to a class of non-autonomous mappings, which can be integrable or not, for which we obtain their exact dynamical degrees. The case of a non-confining singularity is also analysed and again we obtain the corresponding dynamical degrees.

In this paper we develop a bilinearisation-reduction approach to derive solutions to the classical and nonlocal nonlinear Schrödinger (NLS) equations with nonzero backgrounds. We start from the second order Ablowitz-Kaup-Newell-Segur coupled equations as an unreduced system. With a pair of solutions $(q_0,r_0)$ we bilinearize the unreduced system and obtain solutions in terms of quasi double Wronskians. Then we implement reductions by introducing constraints on the column vectors of the Wronskians and finally obtain solutions to the reduced equations, including the classical NLS equation and the nonlocal NLS equations with reverse-space, reverse-time and reverse-space-time, respectively. With a set of plane wave solution $(q_0,r_0)$ as a background solution, we present explicit formulae for these column vectors. As examples, we analyze and illustrate solutions to the focusing NLS equation and the reverse-space nonlocal NLS equation. In particular, we present formulae for the rouge waves of arbitrary order for the focusing NLS equation.