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

In this work, we establish universality results for the $N$-soliton solution of the focusing NLS equation at maximal amplitude. Specifically, we choose the associated normalization constants so that the solution achieves its maximal peak, which, in the large-$N$ limit, satisfies a Painlevé-type equation independently of the distribution of the (random) discrete eigenvalues. We identify two distinct universality classes, determined by the structure of the discrete eigenvalues: the \textit{Painlevé--III} and \textit{Painlevé--V} rogue-wave solutions. In the Painlevé--III case, the eigenvalues take the form $\lambda_j = v_j + i \mu_j$, while for Painlevé--V they satisfy $\lambda_j = -\zeta \, j + v_j + i \mu_j$, with $0 < \zeta < 1$. In both cases, $v_j$ and $\mu_j$ are sub-exponential random variables. Universality can then be summarized as follows: regardless of the specific realizations of the amplitudes and velocities, provided they are sub-exponential random variables and the normalization constants are chosen to maximize the \(N\)-soliton solution, the resulting maximal peak always corresponds to either a Painlevé--III or Painlevé--V rogue-wave profile in the large-$N$ limit.