中国物理B ›› 2024, Vol. 33 ›› Issue (1): 10201-10201.doi: 10.1088/1674-1056/acd686

• •    下一篇

Efficient method to calculate the eigenvalues of the Zakharov—Shabat system

Shikun Cui(崔世坤)1 and Zhen Wang(王振)2,†   

  1. 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;
    2 School of Mathematical Sciences, Beihang University, Beijing 100191, China
  • 收稿日期:2023-02-21 修回日期:2023-05-17 接受日期:2023-05-18 出版日期:2023-12-13 发布日期:2023-12-20
  • 通讯作者: Zhen Wang E-mail:wangzmath@163.com
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 52171251, U2106225, and 52231011) and Dalian Science and Technology Innovation Fund (Grant No. 2022JJ12GX036).

Efficient method to calculate the eigenvalues of the Zakharov—Shabat system

Shikun Cui(崔世坤)1 and Zhen Wang(王振)2,†   

  1. 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;
    2 School of Mathematical Sciences, Beihang University, Beijing 100191, China
  • Received:2023-02-21 Revised:2023-05-17 Accepted:2023-05-18 Online:2023-12-13 Published:2023-12-20
  • Contact: Zhen Wang E-mail:wangzmath@163.com
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 52171251, U2106225, and 52231011) and Dalian Science and Technology Innovation Fund (Grant No. 2022JJ12GX036).

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摘要： A numerical method is proposed to calculate the eigenvalues of the Zakharov—Shabat system based on Chebyshev polynomials. A mapping in the form of (ax) is constructed according to the asymptotic of the potential function for the Zakharov—Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, (ax) mapping, and Chebyshev nodes, the Zakharov—Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma—Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.

关键词: Zakharov—Shabat system, eigenvalue, numerical method, Chebyshev polynomials

Abstract: A numerical method is proposed to calculate the eigenvalues of the Zakharov—Shabat system based on Chebyshev polynomials. A mapping in the form of (ax) is constructed according to the asymptotic of the potential function for the Zakharov—Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, (ax) mapping, and Chebyshev nodes, the Zakharov—Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma—Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.

Key words: Zakharov—Shabat system, eigenvalue, numerical method, Chebyshev polynomials

中图分类号:  (Integrable systems)

  • 02.30.Ik
02.30.Rz (Integral equations) 02.70.Jn (Collocation methods)