Abstract: This paper investigates the asymptotic behavior of high-order vector rogue wave (RW) solutions for any multi-component nonlinear Schrödinger equation (denoted as $n$-NLSE) with multiple internal large parameters. We report some novel RW patterns, including non-multiple root (NMR)-type patterns with distinct shapes such as semicircular sector, acute sector, pseudo-hexagram, and pseudo-rhombus shapes, as well as multiple root (MR)-type patterns characterized by right double-arrow and right arrow shapes. We demonstrate 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 interpreted 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 location corresponding to each simple root of the polynomials and to a lower-order RW at the location 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.

Key words: $n$-NLSE, RW pattern, asymptotics, GMAM polynomial, root structure