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A Riemann-Hilbert Approach to a Generalized Nonlinear Schrödinger Equation on the Quarter Plane

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Abstract

In this work, we investigate a generalized nonlinear Schrödinger equation on the quarter plane. The initial data are vanishing at infinity while the boundary data are time-periodic, of the form ae2iωt+iδ. The main purpose of this work is to consider the long-time asymptotics of the solution to the initial-boundary value problems. Furthermore, we find that the solutions of the initial-boundary value problems have different asymptotics in different regions of the (x,t)-plane. For the region (I) (i.e, \(0<x<4\left (b-\gamma _{1} a^{2}-\sqrt {2}a\right )\)), asymptotics of the solution takes the form of a plane wave. For the region (II) (i.e., \(4\left (b-\gamma _{1} a^{2}-\sqrt {2}a\right )<x<4tb\)), asymptotics of the solution takes the form of a modulated elliptic wave. For the region (III) (i.e., x > 4tb), asymptotics of the solution takes the form of the Zakharov-Manakov vanishing asymptotics.

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Acknowledgments

We express our sincere thanks to the editor and reviewers for their valuable comments. This work is supported by the National Key Research and Development Program of China under Grant No. 2017YFB0202901, the National Natural Science Foundation of China under Grant No.11871180.

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  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Xiu-Bin Wang & Bo Han

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  1. Xiu-Bin Wang
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Correspondence to Xiu-Bin Wang.

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This work is supported by the National Key Research and Development Program of China under Grant No. 2017YFB0202901, the National Natural Science Foundation of China under Grant No.11871180.

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Wang, XB., Han, B. A Riemann-Hilbert Approach to a Generalized Nonlinear Schrödinger Equation on the Quarter Plane. Math Phys Anal Geom 23, 25 (2020). https://doi.org/10.1007/s11040-020-09347-1

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