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Generalized solution to multidimensional cubic Schrödinger equation with delta potential

  • Nevena Dugandžija
  • Marko NedeljkovEmail author
Article
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Abstract

This article addresses the Cauchy problem for the defocusing cubic Schrödinger equation in 2D and 3D and the equation with a delta well potential in 3D. Solutions belong to the Colombeau algebra of generalized functions \(\mathcal {G}_{C^1,H^2}\) (see Nedeljkov et al. in Proc R Soc Edinb Sect A Math 135:863–886, 2005). The physically significant homogeneous problem in 2D and 3D has not yet been treated in this framework, whereas no classical results exist on the equation with delta potential. The paper contains the construction of unique generalized solutions for both of these problems. One could also find an assertion about compatibility with classical solutions for 2D and 3D equations without delta potential.

Keywords

Cubic Schrödinger equation Defocusing equation Colombeau generalized functions 

Mathematics Subject Classification

35Q55 46F30 

Notes

Acknowledgements

The authors are grateful to the referee for extremely valuable comments that make the paper correct and much better. Funding was provided by Ministarstvo Prosvete, Nauke i Tehnološkog Razvoja (RS) (Grant Nos. OI174024, III44006, OI174024) and Provincial Secretariat for Science and Technological Development (Grant No. 114-451-2098/2016).

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Informatics, Faculty of SciencesUniversity of Novi SadNovi SadSerbia

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