Rogue wave patterns associated with Adler–Moser polynomials featuring multiple roots in the nonlinear Schrödinger equation

In this work, we analyze the asymptotic behaviors of high-order rogue wave solutions with multiple large parameters and discover novel rogue wave patterns, including modified claw-like, one triple root (OTR)-type, modified OTR-type, two triple roots (TTR)-type, semimodified TTR-type, and modified TTR-type patterns. A correlation is established between these rogue wave patterns and the root structures of the Adler–Moser polynomials with multiple roots. At the positions in the $(x,t)$-plane corresponding to simple roots of the Adler–Moser polynomials, these high-order rogue wave patterns asymptotically approach first-order rogue waves. At the positions in the $(x,t)$-plane corresponding to multiple roots of the Adler–Moser polynomials, these rogue wave patterns asymptotically tend toward lower-order fundamental rogue waves, dispersed first-order rogue waves, or mixed structures of these rogue waves. These structures are related to the root structures of special Adler–Moser polynomials with new free parameters, such as the Yablonskii–Vorob'ev polynomial hierarchy, among others. Notably, the positions of the fundamental lower-order rogue waves or mixed structures in these rogue wave patterns can be controlled freely under specific conditions.