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Neumann inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation

  • Nakao Hayashi
  • Elena KaikinaEmail author
Article
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Abstract

This paper is the first attempt to give a rigorous mathematical study of Neumann initial boundary value problems for the multidimensional dispersive evolution equations considering as example famous nonlinear Schrödinger equation. We consider the inhomogeneous initial-boundary value problem for the nonlinear Schrödinger equation, formulated on upper right-quarter plane with initial data \(u({\mathbf {x}},t)\left| _{t=0}\right. =u_{0}({\mathbf {x}})\) and Neumann boundary data \(u_{x_{1}}\left| _{\partial _{1}{\mathbf {D}}}\right. =h_{1}(x_{2},t),u_{x_{2}}\left| _{\partial _{2}{\mathbf {D}}}\right. =h_{2}(x_{1},t)\) given in a suitable weighted Lebesgue spaces. We are interested in the study of the influence of the Neumann boundary data on the asymptotic behavior of solutions for large time. We show that problem admits global solutions whose long-time behavior essentially depends on the boundary data. To get a nonlinear theory for the multidimensional model. we propose general method based on Riemann–Hilbert approach and theory Cauchy type integral equations. The advantage of this method is that it can also be applied to non-integrable equations with general inhomogeneous boundary data.

Keywords

Nonlinear Schrödinger equation Large time asymptotics Inhomogeneous Neumann 2D initial-boundary value problem 

Mathematics Subject Classification

Primary 35Q35 
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Notes

Acknowledgements

The work of N.H. is partially supported by JSPS KAKENHI Grant Nos. 25220702, 15H03630. The work of E.I.K. is partially supported by CONACYT and PAPIIT project IN101311.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityOsakaJapan
  2. 2.Centro de Ciencias MatemáticasUNAM Campus MoreliaMoreliaMexico

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