On nonlinear Schrödinger equations with random initial data

Mitia Duerinckx; Mitia Duerinckx

doi:10.3934/mine.2022030

Mathematics in Engineering

2022, Volume 4, Issue 4: 1-14. doi: 10.3934/mine.2022030

Previous Article Next Article

Research article Special Issues

On nonlinear Schrödinger equations with random initial data

Mitia Duerinckx ^{1,2
,
,}

1.
Laboratoire de Mathématiques d'Orsay, CNRS, Université Paris-Saclay, 91400 Orsay, France
2.
Département de Mathématique, Université Libre de Bruxelles, 1050 Brussels, Belgium

Received: 31 July 2021 Accepted: 05 August 2021 Published: 30 August 2021

This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data.
- NLS equations,
- random initial data,
- random fields,
- statistical spatial homogeneity,
- global well-posedness
Citation: Mitia Duerinckx. On nonlinear Schrödinger equations with random initial data[J]. Mathematics in Engineering, 2022, 4(4): 1-14. doi: 10.3934/mine.2022030

Related Papers:

Abstract

This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data.

References

[1]	C. D. Aliprantis, K. C. Border, Infinite dimensional analysis. A hitchhiker's guide, 3 Eds., Berlin: Springer, 2006.
[2]	J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020
[3]	B. Dodson, A. Soffer, T. Spencer, The nonlinear Schrödinger equation on Z and R with bounded initial data: examples and conjectures, J. Stat. Phys., 180 (2020), 910-934. doi: 10.1007/s10955-020-02552-w
[4]	M. Duerinckx, C. Shirley, A new spectral analysis of stationary random Schrödinger operators, J. Math. Phys., 62 (2021), 072106. doi: 10.1063/5.0033583
[5]	J. Ginibre, G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 309-323. doi: 10.1016/S0294-1449(16)30425-5
[6]	J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327. doi: 10.1016/S0294-1449(16)30399-7
[7]	J. Ginibre, G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pure. Appl., 64 (1985), 363-401.
[8]	V. V. Jikov, S. M. Kozlov, O. A. Oleǐnik, Homogenization of differential operators and integral functionals, Berlin: Springer-Verlag, 1994.
[9]	Y. Katznelson, An introduction to harmonic analysis, 3 Eds., Cambridge: Cambridge University Press, 2004.
[10]	S. Nazarenko, Wave turbulence, Heidelberg: Springer, 2011.
[11]	T. Oh, Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data, Commun. Pure Appl. Anal., 14 (2015), 1563-1580. doi: 10.3934/cpaa.2015.14.1563
[12]	T. Oh, On nonlinear Schrödinger equations with almost periodic initial data, SIAM J. Math. Anal., 47 (2015), 1253-1270. doi: 10.1137/140973384
[13]	G. C. Papanicolaou, S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, In: Random fields, Vol. I, II (Esztergom, 1979), North-Holland, Amsterdam, 1981,835-873.
[14]	J. von Neumann, Über Einen Satz Von Herrn M. H. Stone, Ann. Math., 33 (1932), 567-573. doi: 10.2307/1968535

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)