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A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

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Abstract

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation

$$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$$

on the half line (−∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u0(x) = u(x, 0) and the boundary data g0(t) = u(0, t), g1(t) = ux(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.

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Authors and Affiliations

  1. Department of Basical Courses, Shandong University of Science and Technology, Taian, 271019, China

    Ning Zhang

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Ning Zhang & Tie-cheng Xia

  3. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Ning Zhang & En-gui Fan

Authors
  1. Ning Zhang
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  2. Tie-cheng Xia
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  3. En-gui Fan
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Correspondence to Ning Zhang.

Additional information

Supported by the National Natural Science Foundation of China (No.11271008, 61072147, 11671095) and SDUST Research Fund (No.2018TDJH101).

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Zhang, N., Xia, Tc. & Fan, Eg. A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line. Acta Math. Appl. Sin. Engl. Ser. 34, 493–515 (2018). https://doi.org/10.1007/s10255-018-0765-7

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  • DOI: https://doi.org/10.1007/s10255-018-0765-7

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