Painlev\'e Universality classes for the maximal amplitude solution of the Focusing Nonlinear Schr\"{o}dinger Equation with randomness

We establish universality for extremal solutions of the focusing nonlinear Schr\"{o}dinger equation. Extremal solutions are $N$-soliton solutions that achieve the theoretical maximal amplitude and diverge as $N \to \infty$. We consider extremal solutions with the discrete eigenvalues randomly drawn from sub-exponential distributions, and identify two distinct universality classes, determined by the macroscopic structure of the spectrum: the Painlev\'e--III rogue-wave solution, where the eigenvalues take the form $\lambda_j = v_j + i \mu_j$, and the Painlev\'e--V rogue wave solution, where $\lambda_j = -\zeta \, j + v_j + i \mu_j$, with $0 < \zeta < 1$. (In both cases, $\mu_{j}$ and $v_{j}$ are subexponential random variables.) Universality can then be summarized as follows: independently of the specific distribution of the eigenvalues, the rescaled solutions converge locally to a deterministic profile governed by the Painlev\'e-III equation in the first regime, and the Painlev\'e-V equation in the second. These results demonstrate that the formation of Painlev\'e-type rogue waves is a universal phenomenon robust to randomness.