A stochastic thermalization of the discrete nonlinear Schrödinger equation. (English) Zbl 1534.35366
I find this article very interesting. Consider 1d mass subcritical NLS in the periodic case or on torus, in particular the focusing case. It is known that one can construct a natural Gibbs measure, and it can be proven such a Gibbs measure is invariant under the evolution of this equation.
It is however not clear or not expected, in particular in the cubic case, where this model has many/infinity conservation laws, that such a Gibbs measure is the unique invariant measure for the (nonlinear) Schrödinger flow.
One way to overcome this, is to add some noise to the system. The idea of adding noise seems to be natural in those contexts.
But there are two difficulties. 1. Multiplicative noise is hard to study. 2. Additive noise is not mass conserving.
The current article proposes a mass conserving stochastic perturbation of the discrete model, in which case the Gibbs measure becomes the unique invariant distribution. The authors are analyzing this carefully in the article.
It is however not clear or not expected, in particular in the cubic case, where this model has many/infinity conservation laws, that such a Gibbs measure is the unique invariant measure for the (nonlinear) Schrödinger flow.
One way to overcome this, is to add some noise to the system. The idea of adding noise seems to be natural in those contexts.
But there are two difficulties. 1. Multiplicative noise is hard to study. 2. Additive noise is not mass conserving.
The current article proposes a mass conserving stochastic perturbation of the discrete model, in which case the Gibbs measure becomes the unique invariant distribution. The authors are analyzing this carefully in the article.
Reviewer: Chenjie Fan (Beijing)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 39A12 | Discrete version of topics in analysis |
| 60F10 | Large deviations |
| 35C08 | Soliton solutions |
| 35H10 | Hypoelliptic equations |
| 35B20 | Perturbations in context of PDEs |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
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