{"title":"Formal solutions of some family of inhomogeneous nonlinear partial differential equations, Part 2: Summability","authors":"Alberto Lastra , Pascal Remy , Maria Suwińska","doi":"10.1016/j.physd.2024.134420","DOIUrl":null,"url":null,"abstract":"<div><div>In this article, we investigate the summability of the formal power series solutions in time of a class of inhomogeneous nonlinear partial differential equations in two variables, whose corresponding Newton polygon admits a unique positive slope <span><math><mi>k</mi></math></span>, the latter being determined by the highest spatial-derivative order of the initial equation. We give in particular a necessary and sufficient condition for the <span><math><mi>k</mi></math></span>-summability of the solutions in a given direction, and we illustrate this result by some examples. This condition generalizes the ones already given by the second author in Remy (2016, 2020, 2021 [25,26], 2022, 2023). In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof of our main result.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":null,"pages":null},"PeriodicalIF":2.7000,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica D: Nonlinear Phenomena","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167278924003701","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we investigate the summability of the formal power series solutions in time of a class of inhomogeneous nonlinear partial differential equations in two variables, whose corresponding Newton polygon admits a unique positive slope , the latter being determined by the highest spatial-derivative order of the initial equation. We give in particular a necessary and sufficient condition for the -summability of the solutions in a given direction, and we illustrate this result by some examples. This condition generalizes the ones already given by the second author in Remy (2016, 2020, 2021 [25,26], 2022, 2023). In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof of our main result.
在本文中，我们研究了一类两变量非均质非线性偏微分方程的形式幂级数解的时间求和性，其相应的牛顿多边形具有唯一的正斜率 k，后者由初始方程的最高空间衍生阶决定。我们特别给出了在给定方向上解的 k 可求和性的必要条件和充分条件，并通过一些例子说明了这一结果。这个条件概括了第二作者在雷米（2016，2020，2021 [25,26]，2022，2023）中已经给出的条件。此外，我们还介绍了广义二项式系数和多项式系数的一些技术结果，这些结果是证明我们的主要结果所必需的。
期刊介绍：
Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.
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