Broader Universality of Rogue Waves of Infinite Order

We show that the same special solution of the focusing nonlinear Schr\"{o}dinger equation that has been shown to arise in a certain near-field/large-order limit from soliton and Peregrine-like rogue wave solutions actually arises universally from an arbitrary background solution when subjected to a sequence of iterated B\"{a}cklund transformations.Comment: 21 page