Kedziora, David J.Ankiewicz, AdrianAkhmediev, Nail2014-11-062014-11-061539-3755http://hdl.handle.net/1885/12275We present an explicit analytic form for the two-breather solution of the nonlinear Schrödinger equation with imaginary eigenvalues. It describes various nonlinear combinations of Akhmediev breathers and Kuznetsov-Ma solitons. The degenerate case, when the two eigenvalues coincide, is quite involved. The standard inverse scattering technique does not generally provide an answer to this scenario. We show here that the solution can still be found as a special limit of the general second-order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions. A further restriction of this particular case, where the two eigenvalues are equal to i, produces the second-order rogue wave with two free parameters considered as differential shifts. The illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts. Thus we establish a hierarchy of second-order solutions, revealing the interrelated nature of the general case, the rogue wave, and the degenerate breathers.9 pageshttp://www.sherpa.ac.uk/romeo/issn/1539-3755/ "...uthor can archive pre-print (ie pre-refereeing), post-print (ie final draft post-refereeing) and publisher's version/PDF On author's personal website, employer's website or institutional repository. Publisher copyright and source must be acknowledged with citation..." from SHERPA/RoMEO site (as at 6/11/14)Keywords: Degenerate case; Differential shifts; Dinger equation; Eigenvalues; Free parameters; Inverse scattering; Nonlinear combination; Rogue waves; Second orders; Second-order expressions; Second-order solutions; Wave profiles; Eigenvalues and eigenfunctions; Po2012-06-0410.1103/PhysRevE.85.0666012016-02-24