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 November 1, 2024
A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling
As a counterpoint to classical stochastic particle methods for linear diffusion equations, such as Langevin dynamics for the Fokker-Planck equation, we develop a deterministic particle method for the weighted porous medium equation and prove its convergence on bounded time intervals. This generalizes related work on blob methods for unweighted porous medium equations. From a numerical analysis perspective, our method has several advantages: it is meshfree, preserves the gradient flow structure of the underlying PDE, converges in arbitrary dimension, and captures the correct asymptotic behavior in simulations.
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- Award ID(s):
- 2204722
- NSF-PAR ID:
- 10504444
- Publisher / Repository:
- Mathematics of Computation
- Date Published:
- Journal Name:
- Mathematics of Computation
- Volume:
- 92
- Issue:
- 344
- ISSN:
- 0025-5718
- Page Range / eLocation ID:
- 2575 to 2654
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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