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The semi-discrete complex modified Korteweg-de Vries equation with zero and non-zero boundary conditions: Riemann-Hilbert approach and N-soliton solutions

Bo-Jie Deng, Rui Guo, Jian-Wen Zhang

Abstract

We focus on the semi-discrete complex modified Korteweg-de Vries (DcmKdV) equation in this paper. The direct and inverse scattering theory is developed with zero and non-zero boundary conditions (BCs) of the potential. For direct problem, the properties of the eigenfunctions and the scattering matrix, including analyticity, asymptotics and symmetries, are investigated, which facilitates the establishment of the Riemann-Hilbert (RH) problems. By solving the RH problems in the inverse problem part, the reconstruction potential formulas are obtained, which allows us to derive the N-soliton solutions in the reflectionless case. Meanwhile, the trace formulas are derived by means of studying the corresponding RH problems. Furthermore, the dynamic characteristics of the 1-soliton and 2-soliton with zero and non-zero boundary are demonstrated by graphical simulation.

The semi-discrete complex modified Korteweg-de Vries equation with zero and non-zero boundary conditions: Riemann-Hilbert approach and N-soliton solutions

Abstract

We focus on the semi-discrete complex modified Korteweg-de Vries (DcmKdV) equation in this paper. The direct and inverse scattering theory is developed with zero and non-zero boundary conditions (BCs) of the potential. For direct problem, the properties of the eigenfunctions and the scattering matrix, including analyticity, asymptotics and symmetries, are investigated, which facilitates the establishment of the Riemann-Hilbert (RH) problems. By solving the RH problems in the inverse problem part, the reconstruction potential formulas are obtained, which allows us to derive the N-soliton solutions in the reflectionless case. Meanwhile, the trace formulas are derived by means of studying the corresponding RH problems. Furthermore, the dynamic characteristics of the 1-soliton and 2-soliton with zero and non-zero boundary are demonstrated by graphical simulation.
Paper Structure (46 sections, 272 equations, 4 figures)

This paper contains 46 sections, 272 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Zero boundary conditions
  3. Direct scattering problem
  4. Eigenfunctions
  5. Scattering matrix
  6. Summation equations
  7. Analyticity
  8. Proposition 2.1
  9. Proposition 2.2
  10. Asymptotics
  11. Proposition 2.3
  12. Symmetries
  13. Proposition 2.4
  14. Proposition 2.5
  15. Proposition 2.6
  16. ...and 31 more sections

Figures (4)

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Theorems & Definitions (12)

  • proof
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  • proof
  • proof
  • proof
  • proof
  • ...and 2 more