Kedziora, D. J.Ankiewicz, A.Akhmediev, N.2015-09-162015-09-161951-6355http://hdl.handle.net/1885/15440Solutions of the nonlinear Schr¨odinger equation, appearing as rogue waves on a spatially-periodic background envelope, are obtained using the Darboux transformation scheme. Several particular examples are illustrated numerically. These include soliton and breather solutions on a periodic background as well as higher-order structures. The results enrich our knowledge of possible analytic solutions that describe the appearance of rogue waves in a variety of situations.The authors acknowledge the support of the Australian Research Council (Discovery Project number DP110102068). N.A. and A.A. acknowledge support from the Volkswagen Stiftung.© EDP Sciences, Springer-Verlag 2014.2014-01-2310.1140/epjst/e2014-02083-4