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Newton-Based Methods for Finding the Positive Ground State of Gross-Pitaevskii Equations

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Abstract

The discretization of Gross-Pitaevskii equations (GPE) leads to a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). We use two Newton-based methods to compute the positive ground state of GPE. The first method comes from the Newton-Noda iteration for saturable nonlinear Schrödinger equations proposed by Ching-Sung Liu, which can be transferred to GPE naturally. The second method combines the idea of the root-finding methods and the idea of Newton method, in which, each subproblem involving block tridiagonal linear systems can be solved easily. We give an explicit convergence and computational complexity analysis for it. Numerical experiments are provided to support the theoretical results.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions, which help us to improve the paper greatly.

Funding

The first author was funded by the Tianjin Graduate Research and Innovation Project (No. 2019YJSB040). The second author was funded by the National Natural Science Foundation of China (Grant No. 11671217, No. 12071234).

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  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Pengfei Huang & Qingzhi Yang

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  1. Pengfei Huang
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  2. Qingzhi Yang
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Correspondence to Qingzhi Yang.

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This work was funded by the National Natural Science Foundation of China (Grant No. 11671217, No. 12071234) and the Tianjin Graduate Research and Innovation Project (No. 2019YJSB040)

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Cite this article

Huang, P., Yang, Q. Newton-Based Methods for Finding the Positive Ground State of Gross-Pitaevskii Equations. J Sci Comput 90, 49 (2022). https://doi.org/10.1007/s10915-021-01711-9

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