Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions

We present a systematic classification for higher-order rogue-wave solutions of the nonlinear Schrödinger equation, constructed as the nonlinear superposition of first-order breathers via the recursive Darboux transformation scheme. This hierarchy is subdivided into structures that exhibit varying degrees of radial symmetry, all arising from independent degrees of freedom associated with physical translations of component breathers. We reveal the general rules required to produce these fundamental patterns. Consequently, we are able to extrapolate the general shape for rogue-wave solutions beyond order 6, at which point accuracy limitations due to current standards of numerical generation become non-negligible. Furthermore, we indicate how a large set of irregular rogue-wave solutions can be produced by hybridizing these fundamental structures.