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2025
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| Online Access: | https://doi.org/10.5281/zenodo.17837726 |
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper investigates the formation and characteristics of finite-time singularities in nonlinear dispersive partial differential equations (PDEs) subjected to random external driving forces. While deterministic dispersive PDEs can exhibit self-similar blow-up or wave breaking, the introduction of stochastic forcing fundamentally alters the dynamics, leading to a rich interplay between nonlinearity, dispersion, and noise. We explore how random energy injection can either induce, accelerate, or modify the singular behavior, and critically, how it can lead to the emergence of universal scaling laws for these singularities. Through a combination of theoretical analysis, including scaling arguments and statistical mechanics of waves, and extensive numerical simulations of archetypal models such as the Nonlinear Schr"odinger equation, we demonstrate that the characteristics of these noise-induced singularities, such as their amplitude growth rates, spatial profiles, and statistical distributions of collapse times, converge to universal forms largely independent of the specific details of the random forcing or initial conditions. This universality suggests that the interplay of these fundamental mechanisms drives the system towards robust attractors in phase space near collapse. The findings have profound implications for understanding extreme events in various physical systems, ranging from rogue waves in oceans and atmospheric phenomena to optical filamentation and plasma turbulence, where stochasticity plays a crucial role. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17837726 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Emergence of Universal Singularities in Randomly Driven Dispersive Partial Differential Equations Revista, Zen MATH, 10 This paper investigates the formation and characteristics of finite-time singularities in nonlinear dispersive partial differential equations (PDEs) subjected to random external driving forces. While deterministic dispersive PDEs can exhibit self-similar blow-up or wave breaking, the introduction of stochastic forcing fundamentally alters the dynamics, leading to a rich interplay between nonlinearity, dispersion, and noise. We explore how random energy injection can either induce, accelerate, or modify the singular behavior, and critically, how it can lead to the emergence of universal scaling laws for these singularities. Through a combination of theoretical analysis, including scaling arguments and statistical mechanics of waves, and extensive numerical simulations of archetypal models such as the Nonlinear Schr"odinger equation, we demonstrate that the characteristics of these noise-induced singularities, such as their amplitude growth rates, spatial profiles, and statistical distributions of collapse times, converge to universal forms largely independent of the specific details of the random forcing or initial conditions. This universality suggests that the interplay of these fundamental mechanisms drives the system towards robust attractors in phase space near collapse. The findings have profound implications for understanding extreme events in various physical systems, ranging from rogue waves in oceans and atmospheric phenomena to optical filamentation and plasma turbulence, where stochasticity plays a crucial role. |
| title | The Emergence of Universal Singularities in Randomly Driven Dispersive Partial Differential Equations |
| url | https://doi.org/10.5281/zenodo.17837726 |