Initial Boundary Value Problem for the Coupled Kundu Equations on the Half-Line
Abstract
:1. Introduction
2. Basic Riemann–Hilbert Problem
2.1. Formulas and Symbols
- The Pauli matrix can be written as , , , and .
- The matrices S and T are matrices, where S represents the inner product of T; that is, .
- The matrix commutator is , and is calculated in this study as follows: .
- The complex conjugation of a function is represented by an overline, for example, .
2.2. Lax Pair
2.3. Spectral Analysis and Asymptotic Analysis
2.4. Eigenfunctions and Their Relations
2.5. The Spectral Functions and Their Propositions
- (1) .
- (2) The analytic properties of the function are apparent, and .
- (3) The analytic properties of the function are apparent, and .
- (4) The analytic properties of the function are apparent, and .
- (5) The analytic properties of the function are apparent, and .
- (6) The analytic properties of the function are apparent, and .
- (7) The analytic properties of the function are apparent, and .
- (1)
- (2)
- (3)
2.6. Jump Matrix
- (1) contains simple zeros (). We assume that () pertains to , and () pertains to .
- (2) contains simple zeros (). We assume that () pertains to , and () pertains to .
- (3) There are distinctions between the simple zeros of and .
- (1) Res =, .
- (2) Res =, .
- (3) Res =, .
- (4) Res =, .
2.7. The Inverse Problem
3. Definition and Properties of Spectral Functions and Riemann–Hilbert Problem
3.1. Characteristics of Spectral Functions
- (1) For , and are analytical.
- (2) as , .
- (3) As and , the definition of the inverse map for is as follows:
- (1) is a piecewise analytic function.
- (2) , , and
- (3)
- (4) contains simple zeros , (), we assume that () is a part of , and () is a part of .
- (5) There exist simple poles in at , where . The locations of the simple poles in can be identified as , where .
- ,
- ,
- ,
- ,
- (1) For , and are analytical.
- (2) as , .
- (3) As and , the definition of the inverse map for is as follows:
- (1) is a piecewise analytic function.
- (2) , , and
- (3)
- (4) contains simple zeros , (), and we assume that () is a part of and () is a part of .
- (5) There exist simple poles in at , where . The locations of the simple poles in can be identified as , where .
- (1) For , and are analytical;
- (2) as , ;
- (3) , ;
- (4) , ;
- (5) As and , the definition of the inverse map for is as follows:
- (1) is a piecewise analytic function.
- (2) , , and
- (3)
- (4) contains simple zeros (), we assume that () is a part of , and () is a part of .
- (5) There exist simple poles in at , where . The locations of the simple poles in can be identified as , where .
3.2. Riemann–Hilbert Problem
- The function is an analytic function that operates on slices while maintaining a determinant of unity.
- meets the jump condition
- Simple poles are located at , , and , in . Additionally, simple poles exist at , , and , in .
- .
- fulfills the residue relationship mentioned in Hypothesis 1.
- (1)
- (2)
4. Conclusions and Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Hu, J.; Zhang, N. Initial Boundary Value Problem for the Coupled Kundu Equations on the Half-Line. Axioms 2024, 13, 579. https://doi.org/10.3390/axioms13090579
Hu J, Zhang N. Initial Boundary Value Problem for the Coupled Kundu Equations on the Half-Line. Axioms. 2024; 13(9):579. https://doi.org/10.3390/axioms13090579
Chicago/Turabian StyleHu, Jiawei, and Ning Zhang. 2024. "Initial Boundary Value Problem for the Coupled Kundu Equations on the Half-Line" Axioms 13, no. 9: 579. https://doi.org/10.3390/axioms13090579