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

  • Nakao Hayashi
  • Elena KaikinaEmail author


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.


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

Mathematics Subject Classification

Primary 35Q35 



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.


  1. 1.
    Barab, J.E.: Non-existence of asymptotically free solutions for nonlinear Schrödinger equation. J. Math. Phys. 25(11), 3270–3273 (1984)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bu, Q.: On well-posedness of the forced nonlinear Schrödinger equation. Appl. Anal. 46(3–4), 219–239 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bu, C.Q., Strauss, W.: An inhomogeneous boundary value problem for nonlinear Schrodinger equations. J. Differ. Equ. 173, 79–91 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cazenave, Th: Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences, New York, p. xiv+323. American Mathematical Society, Providence (2003)Google Scholar
  5. 5.
    Fokas, A.S., Its, A.R., Sung, L.-Y.: The nonlinear Schrödinger equation on the half-line. Nonlinearity 18(4), 1771–1822 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hayashi, N., Kaikina, E.: Nonlinear Theory of Pseudodifferential Equations on a Half-Line. North-Holland Mathematics Studies, vol. 194, p. 319. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  7. 7.
    Hayashi, N., Naumkin, P.I.: Asymptotics of small solutions to nonlinear Schrödinger equation with cubic nonlinearities. Int. J. Pure Appl. Math. 3, 255–273 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Holmer, J.: The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line. Differ. Integral Equ. 18(6), 647–668 (2005)zbMATHGoogle Scholar
  9. 9.
    Kaikina, E.I.: A new unified approach to study fractional PDE equations on a half-line. Complex Var. Elliptic Equ. 58(1), 55–77 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kaikina, E.: Asymptotics for inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation. J. Math. Phys. 54(11), 111504 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kaikina, E.I.: Inhomogeneous Neumann initial boundary value problem for the nonlinear Schrödinger equation. J. Differ. Equ. 255, 33383356 (2013)CrossRefGoogle Scholar
  12. 12.
    Kaikina, E.: Inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation. J. Math. Phys. 59(6), 061506 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kaikina, E.I.: Forced cubic Schrödinger equation with Robin boundary data: large-time asymptotics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469(2159), 20130341 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Monvel, A., Kotlyarov, V., Shepelsky, D.: Decaying long-time asymptotics for the focusing NLS equation with periodic boundary condition. Stud. Appl. Math. 129(3), 249–271 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Naumkin, I.P.: Sharp asymptotic behavior of solutions for cubic nonlinear Schrödinger equations with a potential. J. Math. Phys. 57(5), 051501 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Naumkin, I.P.: Cubic nonlinear Dirac equation in a quarter plane. J. Math. Anal. Appl. 434(2), 1633–1664 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Naumkin, I.P.: Klein-Gordon equation with critical nonlinearity and inhomogeneous Dirichlet boundary conditions. Differ. Integral Equ. 29(1–2), 55–92 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Naumkin, I.P.: Initial-boundary value problem for the one dimensional Thirring model. J. Differ. Equ. 261(8), 4486–4523 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ozawa, T.: Long range scattering for nonlinear Schrödinger equations in one space dimension. Commun. Math. Phys. 139, 479–493 (1991)CrossRefGoogle Scholar

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© 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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