{"title":"Global Existence and Long-Time Behavior in the 1 + 1-Dimensional Principal Chiral Model with Applications to Solitons","authors":"Jessica Trespalacios","doi":"10.1007/s00023-023-01405-y","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider the 1 + 1-dimensional vector-valued principal chiral field model (PCF) obtained as a simplification of the vacuum Einstein field equations under the Belinski–Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step toward the study of its nonlinear stability.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"25 11","pages":"4671 - 4712"},"PeriodicalIF":1.4000,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales Henri Poincaré","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1007/s00023-023-01405-y","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we consider the 1 + 1-dimensional vector-valued principal chiral field model (PCF) obtained as a simplification of the vacuum Einstein field equations under the Belinski–Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step toward the study of its nonlinear stability.
期刊介绍：
The two journals Annales de l''Institut Henri Poincaré, physique théorique and Helvetica Physical Acta merged into a single new journal under the name Annales Henri Poincaré - A Journal of Theoretical and Mathematical Physics edited jointly by the Institut Henri Poincaré and by the Swiss Physical Society.
The goal of the journal is to serve the international scientific community in theoretical and mathematical physics by collecting and publishing original research papers meeting the highest professional standards in the field. The emphasis will be on analytical theoretical and mathematical physics in a broad sense.
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