arXiv subjects

Huian Lin

Publications and source records attributed to Huian Lin.

3 recordsLinked to original sources

arXiv2026

Coexistence of two distinct rogue wave patterns in the coupled nonlinear Schr\"odinger equation

This paper investigates the asymptotic behavior of high-order vector rogue wave (RW) solutions of the coupled nonlinear Schr\"odinger (CNLS) equation in the presence of multiple large internal parameters. We report several new high-order RW patterns in the CNLS system, including double-sector, double-heart, and mixed sector-heart configurations. The main novelty is that each RW pattern contains two distinct regions in which two different fundamental first-order RWs coexist simultaneously, potentially appearing as bright (eye-shaped) versus four-petaled or dark (anti-eye-shaped) forms. These two regions are respectively associated with the simple root structures of two different Adler--Moser polynomials: each region consists of well-separated first-order RWs in one-to-one correspondence with the simple roots of the associated polynomial. In addition, by tuning certain free parameters, the two regions of the RW pattern can be shifted to arbitrary locations in the $ (x,t) $-plane. This flexibility, together with the rich simple-root structures of Adler--Moser polynomials, enables the systematic generation of a much broader family of structured RW patterns in the CNLS equation.

nlin.SI↗

arXiv2025

Vector rogue wave patterns of the multi-component nonlinear Schr\"odinger equation and generalized mixed Adler--Moser polynomials

This paper investigates the asymptotic behavior of high-order vector rogue wave (RW) solutions for any multi-component nonlinear Schr\"odinger equation (denoted as $n$-NLSE) with multiple internal large parameters and reports some new RW patterns, including non-multiple root (NMR)-type patterns with shapes such as $ 180 $-degree sector, jellyfish-like, and thumbtack-like shapes, as well as multiple root (MR)-type patterns characterized by right double-arrow and right arrow shapes. We establish that these RW patterns are intrinsically related to the root structures of a novel class of polynomials, termed generalized mixed Adler--Moser (GMAM) polynomials, which feature multiple arbitrary free parameters. The RW patterns can be understood as straightforward expansions and slight shifts of the root structures for the GMAM polynomials to some extent. In the $(x,t)$-plane, they asymptotically converge to a first-order RW at the position corresponding to each simple root of the polynomials and to a lower-order RW at the position associated with each multiple root. Notably, the position of the lower-order RW within these patterns can be flexibly adjusted to any desired location in the $(x,t)$-plane by tuning the free parameters of the corresponding GMAM polynomials.

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arXiv2024

Rogue wave patterns associated with Adler--Moser polynomials featuring multiple roots in the nonlinear Schr\"odinger 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 claw-like, OTR-type, TTR-type, semi-modified TTR-type, and their modified 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 single 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.

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