PreprintCC BY 4.0arXiv
2D Torus defocusing NLSE with a potential and random initial data
Abstract
We consider the non-linear cubic defocusing Schrödinger equation on the 2D Torus (T2) with a potential V(x)∈ H2+(T2). The initial data, u0,is given by a centered gaussian variable with covariance (−Δ+V)−1, and thus is almost surely in H0-(T2). We show that it is almost surely well-possed for all time in the weak sense, and that u−eit(Δ−V)u0 is in Hs(T2) for some s>0, and that the solution is the limit of the solutions to the truncated equation. This result is a generalization of the results first given by Bourgain in Invariant Measures for the {2D}-Defocusing Nonlinear {Schr{ö}dinger} Equation.
Related
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BibTeX
@misc{lindstrom2026torus,
title = {{2D Torus defocusing NLSE with a potential and random initial data}},
author = {Nicolas Lindstrom},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21969},
eprint = {2609.21969},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21969},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - 2D Torus defocusing NLSE with a potential and random initial data
AU - Nicolas Lindstrom
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21969
UR - https://arxiv.org/abs/2609.21969
AB - We consider the non-linear cubic defocusing Schrödinger equation on the 2D Torus (T^2) with a potential V(x)∈ H^2^+(T^2). The initial data, u_0,is given by a centered gaussian variable with covariance (−Δ+V)^−1, and thus is almost surely in H^0^-(T^2). We show that it is almost surely well-possed for all time in the weak sense, and that u−e^it(Δ−V)u_0 is in H^s(T^2) for some s>0, and that the solution is the limit of the solutions to the truncated equation. This result is a generalization of the results first given by Bourgain in Invariant Measures for the {2D}-Defocusing Nonlinear {Schr{ö}dinger} Equation.
ER -