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Maillet type theorem for nonlinear totally characteristic partial differential equations

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Abstract

The paper discusses a holomorphic nonlinear singular partial differential equation \((t \partial _t)^mu=F(t,x,\{(t \partial _t)^j \partial _x^{\alpha }u \}_{j+\alpha \le m, j<m})\) under the assumption that the equation is of nonlinear totally characteristic type. By using the Newton polygon at \(x=0\), the notion of the irregularity at \(x=0\) of the equation is defined. In the case where the irregularity is greater than one, it is proved that every formal power series solution belongs to a suitable formal Gevrey class. The precise bound of the order of the formal Gevrey class is given, and the optimality of this bound is also proved in a generic case.

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Acknowledgements

The authors would like to thank the anonymous referee who has pointed out the illustrative example described in the Remark in Sect. 2.4.

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Authors and Affiliations

  1. Departamento de Física y Matemáticas, University of Alcalá, Ap. de Correos 20, 28871, Alcalá de Henares, Madrid, Spain

    Alberto Lastra

  2. Department of Information and Communication Sciences, Sophia University, Kioicho, Chiyoda-ku, Tokyo, 102-8554, Japan

    Hidetoshi Tahara

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  1. Alberto Lastra
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Correspondence to Alberto Lastra.

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Communicated by Y. Giga.

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A. Lastra is partially supported by the project MTM2016-77642-C2-1-P of Ministerio de Economía y Competitividad, Spain. H. Tahara is partially supported by JSPS KAKENHI Grant number 15K04966.

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Lastra, A., Tahara, H. Maillet type theorem for nonlinear totally characteristic partial differential equations. Math. Ann. 377, 1603–1641 (2020). https://doi.org/10.1007/s00208-019-01864-x

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  • DOI: https://doi.org/10.1007/s00208-019-01864-x

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