TY - JOUR

T1 - Neumann inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation

AU - Hayashi, Nakao

AU - Kaikina, Elena

N1 - Funding Information:
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.
Funding Information:
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. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Publisher Copyright:
© 2019, Springer Nature Switzerland AG.

PY - 2020/2/1

Y1 - 2020/2/1

N2 - 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(x, t) |t == u(x) and Neumann boundary data ux1|∂1D=h1(x2,t),ux2|∂2D=h2(x1,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.

AB - 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(x, t) |t == u(x) and Neumann boundary data ux1|∂1D=h1(x2,t),ux2|∂2D=h2(x1,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.

KW - Inhomogeneous Neumann 2D initial-boundary value problem

KW - Large time asymptotics

KW - Nonlinear Schrödinger equation

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U2 - 10.1007/s00030-019-0605-3

DO - 10.1007/s00030-019-0605-3

M3 - Article

AN - SCOPUS:85075469527

SN - 1021-9722

VL - 27

JO - Nonlinear Differential Equations and Applications

JF - Nonlinear Differential Equations and Applications

IS - 1

M1 - 2

ER -