Triangular rogue clusters associated with multiple roots of Adler--Moser polynomials in integrable systems

Rogue patterns associated with multiple roots of Adler--Moser polynomials under general multiple large parameters are studied in integrable systems. It is first shown that the multiplicity of any multiple root in any Adler--Moser polynomial is a triangular number (i.e., its multiplicity is equal to $n(n+1)/2$ for a certain integer $n$). Then, it is shown that corresponding to a nonzero multiple root of the Adler--Moser polynomial, a triangular rogue cluster would appear on the spatial-temporal plane. This triangular rogue cluster comprises $n(n+1)/2$ fundamental rogue waves forming a triangular shape, and space-time locations of fundamental rogue waves in this triangle are a linear transformation of the Yablonskii--Vorob'ev polynomial $Q_{n}(z)$'s root structure. In the special case where this multiple root of the Adler--Moser polynomial is zero, the associated rogue pattern is found to be a $n$-th order rogue wave in the $O(1)$ neighborhood of the spatial-temporal origin. These general results are demonstrated on two integrable systems: the nonlinear Schr\"odinger equation and the generalized derivative nonlinear Schr\"odinger equation. For these equations, asymptotic predictions of rogue patterns are compared with true rogue solutions and good agreement between them is illustrated.