Rigorous Computation of Solutions of Semilinear PDEs on Unbounded Domains via Spectral Methods

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-22 DOI:10.1137/23m1607507

Matthieu Cadiot, Jean-Philippe Lessard, Jean-Christophe Nave

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引用次数: 0

Abstract

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1966-2017, September 2024.
Abstract.In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semilinear PDEs in a Hilbert space [math] ([math]) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in [math] as well as bounded linear operators from [math] to [math]. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton–Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters, such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in [math] as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.

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通过谱方法严格计算无界域上半线性 PDE 的解

SIAM 应用动力系统期刊》，第 23 卷第 3 期，第 1966-2017 页，2024 年 9 月。摘要.本文提出了一种通用方法，通过计算机辅助证明，严格证明希尔伯特空间[math]（[math]）中一大类自治半线性 PDE 的强解的存在性。我们的方法是全谱性的，使用傅立叶级数来近似 [math] 中的函数以及 [math] 到 [math] 中的有界线性算子。特别是，我们通过傅里叶级数近似来构造微分算子的近似逆。将这种构造与牛顿-康托洛维奇方法相结合，我们开发了一种数值方法来证明强解的存在性。为此，我们引入了有限维迹线定理，并据此建立了在超立方体上有支持的平滑函数。然后，我们将该方法推广到具有额外方程/参数的 PDE 系统，如特征值问题。作为应用，我们证明了[math]中川原方程中存在行波（孤子），以及关于孤子线性化的特征对。通过这些结果，我们可以证明上述行波的稳定性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.