Determine the NLS rogue-wave pattern induced by a multiple root of the Adler–Moser polynomial under single-power multiple large parameters

Determine the spatial–temporal wave pattern induced by a multiple root of the Adler–Moser polynomial Θ_N(z; κ_1, …, κ_{N−1}) in the nonlinear Schrödinger equation’s rogue-wave solutions u_N(x, t) when the internal parameters are of the single-power multiple-large form a_{2j+1} = κ_j A^{2j+1} for 1 ≤ j ≤ N−1 with A ≫ 1 and arbitrary O(1) complex coefficients κ_j.

Background

Earlier work established that when all roots of the Adler–Moser polynomial are simple, each simple root induces a fundamental Peregrine rogue wave whose location is linearly related to the root. However, the case where the Adler–Moser polynomial admits multiple roots had not been resolved in prior studies focusing on single-power multiple large parameters.

The authors note that Lin and Ling (2025) considered multiple-root patterns under a more involved form of large parameters with special coefficient values that do not cover the single-power form a_{2j+1} = κ_j A^{2j+1} with arbitrary coefficients. This left the question of the pattern induced by a multiple root under the single-power form open at that point in the narrative.