The solutions of classical and nonlocal nonlinear Schr\"{o}dinger equations with nonzero backgrounds: Bilinearisation and reduction approach

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