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Response Solutions for KdV Equations with Liouvillean Frequency

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Abstract

In this paper, we prove an infinite dimensional KAM (Kolmogorov–Arnold–Moser) theorem, which can be used to the KdV equations

$${u_t} + {\partial _{xxx}}u - \varepsilon {\partial _x}f(\omega t,x,u) = 0,$$

where \(\omega = \xi \bar \omega ,\,\,\bar \omega = (1,\alpha)\) is Liouvillean forced frequency and f is real analytic. We obtain a C∞ smooth response solution under zero mean-value periodic boundary conditions. The proof is based on a modified infinite dimensional KAM theory.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11971012). Yingnan Sun was supported by CSC (No. 202006190134).

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Authors and Affiliations

  1. Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, China

    Ningning Chang

  2. School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Yingnan Sun

  3. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Jiansheng Geng & Yingnan Sun

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  1. Ningning Chang
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  2. Jiansheng Geng
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Correspondence to Jiansheng Geng.

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Chang, N., Geng, J. & Sun, Y. Response Solutions for KdV Equations with Liouvillean Frequency. Front. Math 18, 1083–1112 (2023). https://doi.org/10.1007/s11464-021-0099-2

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  • DOI: https://doi.org/10.1007/s11464-021-0099-2

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