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Efficient method to calculate the eigenvalues of the Zakharov—Shabat system |
| Shikun Cui(崔世坤)1 and Zhen Wang(王振)2,† |
1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;
2 School of Mathematical Sciences, Beihang University, Beijing 100191, China |
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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.
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Received: 21 February 2023
Revised: 17 May 2023
Accepted manuscript online: 18 May 2023
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PACS:
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02.30.Ik
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(Integrable systems)
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02.30.Rz
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(Integral equations)
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02.70.Jn
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(Collocation methods)
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| Fund: 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). |
Corresponding Authors:
Zhen Wang
E-mail: wangzmath@163.com
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Cite this article:
Shikun Cui(崔世坤) and Zhen Wang(王振) Efficient method to calculate the eigenvalues of the Zakharov—Shabat system 2024 Chin. Phys. B 33 010201
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