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The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach

Author

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  • Tongshuai Liu

    (College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China)

  • Huanhe Dong

    (College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China)

Abstract

In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formulated in the complex plane with respect to the spectral parameter. According to the initial-boundary values, the spectral function can be defined. Furthermore, the jump matrices and the global relations can be obtained. Finally, the potential q ( x , t ) can be represented by the solution of this Riemann-Hilbert problem.

Suggested Citation

  • Tongshuai Liu & Huanhe Dong, 2019. "The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach," Mathematics, MDPI, vol. 7(2), pages 1-17, February.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:2:p:170-:d:205678
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    References listed on IDEAS

    as
    1. Mengshuang Tao & Huanhe Dong, 2017. "Algebro-Geometric Solutions for a Discrete Integrable Equation," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-9, November.
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    4. Wen-Xiu Ma & Jie Li & Chaudry Masood Khalique, 2018. "A Study on Lump Solutions to a Generalized Hirota-Satsuma-Ito Equation in (2+1)-Dimensions," Complexity, Hindawi, vol. 2018, pages 1-7, December.
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    6. Changna Lu & Qianqian Gao & Chen Fu & Hongwei Yang, 2017. "Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-11, November.
    7. Hu, Bei-Bei & Xia, Tie-Cheng & Ma, Wen-Xiu, 2018. "Riemann–Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 148-159.
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