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A prototypical example of a rogue wave structure in a two-dimensional modelis presented in the context of the Davey-Stewartson~II (DS~II) equation arisingin water waves. The analytical methodology involves a Taylor expansion of aneigenfunctionof the model's Lax pair which is used to form a hierarchy ofinfinitely many new eigenfunctions. These are used for the construction oftwo-dimensional (2D) rogue waves (RWs) of the DS~II equation by the even-foldDarboux transformation (DT). The obtained 2D RWs, which are localized in bothspace and time, can be viewed as a 2D analogue of the Peregrine soliton and arethus natural candidates to describe oceanic RW phenomena,as well as ones in 2Dfluid systems and water tanks.

Two-dimensional rogue waves on zero background of the Davey-Stewartson II equation