Painlevé Universality classes for the maximal amplitude solution of the Focusing Nonlinear Schrödinger Equation with randomness

We establish universality for extremal solutions of the focusing nonlinear Schrödinger equation. Extremal solutions are $N$-soliton solutions that achieve the theoretical maximal amplitude and diverge as $N\to \mathrm{\infty }$. 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 Painlevé–III rogue-wave solution, where the eigenvalues take the form ${\lambda }_j=v_j+i{\mu }_j$, and the Painlevé–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 Painlevé-III equation in the first regime, and the Painlevé-V equation in the second. These results demonstrate that the formation of Painlevé-type rogue waves is a universal phenomenon robust to randomness.