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Rogue wave patterns in the nonlinear Schrödinger equation

Rogue wave patterns in the nonlinear Schrödinger equation are analytically studied. It is shown that when an internal parameter in the rogue waves (which controls the shape of initial weak perturbations to the uniform background) is large, these waves would exhibit clear geometric structures, which are formed by Peregrine waves in shapes such as triangle, pentagon, heptagon and nonagon, with a possible lower-order rogue wave at its center. These rogue patterns are analytically determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy, and their orientations are controlled by the phase of the large parameter. It is also shown that when multiple internal parameters in the rogue waves are large but satisfy certain constraints, similar rogue patterns would still hold. Comparison between true rogue patterns and our analytical predictions shows excellent agreement.