Painlevé Universality classes for the maximal amplitude solution of the Focusing Nonlinear Schrödinger Equation with randomness
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908814099349504 |
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| author | Gkogkou, Aikaterini Mazzuca, Guido McLaughlin, Kenneth D. T-R |
| author_facet | Gkogkou, Aikaterini Mazzuca, Guido McLaughlin, Kenneth D. T-R |
| contents | 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 $N \to \infty$. 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 $λ_j = v_j + i μ_j$, and the Painlevé--V rogue wave solution, where $λ_j = -ζ\, j + v_j + i μ_j$, with $0 < ζ< 1$. (In both cases, $μ_{j}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05101 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Painlevé Universality classes for the maximal amplitude solution of the Focusing Nonlinear Schrödinger Equation with randomness Gkogkou, Aikaterini Mazzuca, Guido McLaughlin, Kenneth D. T-R Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems 35Q55 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 $N \to \infty$. 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 $λ_j = v_j + i μ_j$, and the Painlevé--V rogue wave solution, where $λ_j = -ζ\, j + v_j + i μ_j$, with $0 < ζ< 1$. (In both cases, $μ_{j}$ 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. |
| title | Painlevé Universality classes for the maximal amplitude solution of the Focusing Nonlinear Schrödinger Equation with randomness |
| topic | Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems 35Q55 |
| url | https://arxiv.org/abs/2602.05101 |