{"title":"Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlevé IV transition","authors":"Zhi-Xuan Meng , Shuai-Xia Xu , Yu-Qiu Zhao","doi":"10.1016/j.physd.2025.134589","DOIUrl":null,"url":null,"abstract":"<div><div>In the present paper, we study the time-dependent correlation function of the one-dimensional impenetrable Bose gas, which can be expressed in terms of the Fredholm determinant of a time-dependent sine kernel and the solutions of the separated NLS equations. We derive the large time and distance asymptotic expansions of this determinant and the solutions of the separated NLS equations in both the space-like region and time-like region of the <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span>-plane. Furthermore, we observe a phase transition between the asymptotic expansions in these two different regions. The phase transition is then shown to be described by a particular solution of the Painlevé IV equation.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134589"},"PeriodicalIF":2.7000,"publicationDate":"2025-02-27","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/S0167278925000685","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
摘要
本文研究了一维不可穿透玻色气体的时相关函数,它可以用时相关正弦核的弗雷德霍姆行列式和分离 NLS 方程的解来表示。我们推导了该行列式和分离 NLS 方程在 (x,t) 平面的类空间区域和类时间区域的大时间和距离渐近展开。此外,我们还观察到这两个不同区域的渐近展开之间存在相变。相变可以用潘列韦 IV 方程的一个特定解来描述。
Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlevé IV transition
In the present paper, we study the time-dependent correlation function of the one-dimensional impenetrable Bose gas, which can be expressed in terms of the Fredholm determinant of a time-dependent sine kernel and the solutions of the separated NLS equations. We derive the large time and distance asymptotic expansions of this determinant and the solutions of the separated NLS equations in both the space-like region and time-like region of the -plane. Furthermore, we observe a phase transition between the asymptotic expansions in these two different regions. The phase transition is then shown to be described by a particular solution of the Painlevé IV equation.
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.