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

Abstract

In this paper, novel rogue wave patterns in the nolocal nonlinear Schr\"odinger equation (NLS) are investigated by means of asymptotic analysis, including heart-pentagon, oval-trangle, and fan-trangle. It is demonstrated that when multiple free parameters get considerably large, rogue wave patterns can approximately be predicted by the root structures of Adler-Moser polynomials. These polynomials, which extend the Yablonskii-Vorob'ev polynomial hierarchy, exhibit richer geometric shapes in their root distributions. The (x,t)-plane is partitioned into three regions and through a combination of asymptotic results in different regions, unreported rogue wave patterns can be probed. Predicted solutions are compared with true rogue waves in light of graphical illustrations and numerical confirmation, which reveal excellent agreement between them.