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Riemann–Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line

Author

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  • Hu, Bei-Bei
  • Xia, Tie-Cheng
  • Ma, Wen-Xiu

Abstract

In this work, we investigate the two-component modified Korteweg-de Vries (mKdV) equation, which is a complete integrable system, and accepts a generalization of 4 × 4 matrix Ablowitz–Kaup–Newell-Segur (AKNS)-type Lax pair. By using of the unified transform approach, the initial-boundary value (IBV) problem of the two-component mKdV equation associated with a 4 × 4 matrix Lax pair on the half-line will be analyzed. Supposing that the solution {u1(x, t), u2(x, t)} of the two-component mKdV equation exists, we will show that it can be expressed in terms of the unique solution of a 4 × 4 matrix Riemann–Hilbert problem formulated in the complex λ-plane. Moreover, we will prove that some spectral functions s(λ) and S(λ) are not independent of each other but meet the global relationship.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:332:y:2018:i:c:p:148-159
    DOI: 10.1016/j.amc.2018.03.049
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    Cited by:

    1. Hu, Beibei & Zhang, Ling & Xia, Tiecheng & Zhang, Ning, 2020. "On the Riemann-Hilbert problem of the Kundu equation," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    2. 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.
    3. Guan, Xue & Liu, Wenjun & Zhou, Qin & Biswas, Anjan, 2020. "Some lump solutions for a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation," Applied Mathematics and Computation, Elsevier, vol. 366(C).
    4. Hu, Beibei & Lin, Ji & Zhang, Ling, 2022. "On the Riemann-Hilbert problem for the integrable three-coupled Hirota system with a 4×4 matrix Lax pair," Applied Mathematics and Computation, Elsevier, vol. 428(C).
    5. Bo Xu & Sheng Zhang, 2022. "Analytical Method for Generalized Nonlinear Schrödinger Equation with Time-Varying Coefficients: Lax Representation, Riemann-Hilbert Problem Solutions," Mathematics, MDPI, vol. 10(7), pages 1-15, March.
    6. Liu, Ling & Wen, Xiao-Yong & Liu, Nan & Jiang, Tao & Yuan, Jin-Yun, 2020. "An integrable lattice hierarchy associated with a 4 × 4 matrix spectral problem: N-fold Darboux transformation and dynamical properties," Applied Mathematics and Computation, Elsevier, vol. 387(C).

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