Rogue wave patterns associated with Adler-Moser polynomials in the nonlinear Schr\"odinger equation

Abstract

We report new rogue wave patterns in the nonlinear Schr\"{o}dinger equation. These patterns include heart-shaped structures, fan-shaped sectors, and many others, that are formed by individual Peregrine waves. They appear when multiple internal parameters in the rogue wave solutions get large. Analytically, we show that these new patterns are described asymptotically by root structures of Adler-Moser polynomials through a dilation. Since Adler-Moser polynomials are generalizations of the Yablonskii-Vorob'ev polynomial hierarchy and contain free complex parameters, these new rogue patterns associated with Adler-Moser polynomials are much more diverse than previous rogue patterns associated with the Yablonskii-Vorob'ev polynomial hierarchy. We also compare analytical predictions of these patterns to true solutions and demonstrate good agreement between them.