Complex-valued solutions of the mKdV equations in generalized Fourier-Lebesgue spaces

Chen, Zijun; Guo, Zihua; Huang, Chunyan

doi:10.1090/proc/17114

Complex-valued solutions of the mKdV equations in generalized Fourier-Lebesgue spaces
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by Zijun Chen, Zihua Guo and Chunyan Huang;

Proc. Amer. Math. Soc. 153 (2025), 1621-1640

DOI: https://doi.org/10.1090/proc/17114

Published electronically: February 7, 2025

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Abstract:

We study the complex-valued solutions to the Cauchy problem of the modified Korteweg-de Vries equation on the real line. To study the low-regularity problems, we introduce a generalized Fourier-Lebesgue space $\widehat {M}^{s}_{r,q}(\mathbb {R})$ that unifies the modulation spaces and the Fourier-Lebesgue spaces. We then prove sharp local well-posedness results in this space by perturbation arguments using $X^{s,b}$-type spaces. Our results improve the previous one in Grünrock and Vega [Trans. Amer. Math. Soc. 361 (2009), pp. 5681–5694].

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