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We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread with time, and this fact answers the “weak turbulence” question for the nonlinear Schrödinger equation with random potentials. We also derive Ohm’s law for the porous medium equation.
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This content will become publicly available on June 1, 2026
Numerical schemes for a fully nonlinear coagulation–fragmentation model coming from wave kinetic theory
This article introduces a novel numerical approach, based on finite-volume techniques, for studying fully nonlinear coagulation–fragmentation models, where both the coagulation and fragmentation components of the collision operator are nonlinear. The models come from three-wave kinetic equations, a pivotal framework in wave turbulence theory. Despite the importance of wave turbulence theory in physics and mechanics, there have been very few numerical schemes for three-wave kinetic equations, in which no additional assumptions are manually imposed on the evolution of the solutions, and the current manuscript provides one of the first of such schemes. To the best of our knowledge, this also is the first numerical scheme capable of accurately capturing the long-term asymptotic behaviour of solutions to a fully nonlinear coagulation–fragmentation model. The scheme is implemented on some test problems, demonstrating strong alignment with theoretical predictions of energy cascade rates, rigorously obtained in the work (Soffer & Tran. 2020Commun. Math. Phys.376, 2229–2276. (doi:10.1007/BF01419532)). We further introduce a weighted finite-volume variant to ensure energy conservation across varying degrees of kernel homogeneity. Convergence and first-order consistency are established through theoretical analysis and verified by experimental convergence orders in test cases.
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- PAR ID:
- 10610428
- Publisher / Repository:
- Royal Society
- Date Published:
- Journal Name:
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 481
- Issue:
- 2316
- ISSN:
- 1364-5021
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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