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Soliton resolution for the generalized complex short pulse equation with the weighted Sobolev initial data 具有加权Sobolev初始数据的广义复短脉冲方程的孤子解析
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-28 DOI: 10.1016/j.geomphys.2024.105387
Xianguo Geng , Feiying Yan , Jiao Wei
{"title":"Soliton resolution for the generalized complex short pulse equation with the weighted Sobolev initial data","authors":"Xianguo Geng ,&nbsp;Feiying Yan ,&nbsp;Jiao Wei","doi":"10.1016/j.geomphys.2024.105387","DOIUrl":"10.1016/j.geomphys.2024.105387","url":null,"abstract":"<div><div>In this work, the Cauchy problem for the generalized complex short pulse equation with initial conditions in the weighted Sobolev space <span><math><mi>H</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is studied by using the Riemann-Hilbert method and the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-steepest descent method. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem can be expressed as solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, the long-time asymptotics and soliton resolution of the generalized complex short pulse equation in the soliton region are obtained by resorting to the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-steepest descent method. The results also indicate that the <em>N</em>-soliton solutions of the generalized complex short pulse equation are asymptotically stable.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105387"},"PeriodicalIF":1.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142747797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the third coefficient in the TYCZ–expansion of the epsilon function of Kähler–Einstein manifolds 论卡勒-爱因斯坦流形ε函数 TYCZ 展开中的第三个系数
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-26 DOI: 10.1016/j.geomphys.2024.105384
Simone Cristofori, Michela Zedda
{"title":"On the third coefficient in the TYCZ–expansion of the epsilon function of Kähler–Einstein manifolds","authors":"Simone Cristofori,&nbsp;Michela Zedda","doi":"10.1016/j.geomphys.2024.105384","DOIUrl":"10.1016/j.geomphys.2024.105384","url":null,"abstract":"<div><div>In this paper we compute the third coefficient arising from the TYCZ-expansion of the <em>ε</em>-function associated to a Kähler-Einstein metric and discuss the consequences of its vanishing.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105384"},"PeriodicalIF":1.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The constraint tensor for null hypersurfaces 空超曲面的约束张量
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-22 DOI: 10.1016/j.geomphys.2024.105375
Miguel Manzano, Marc Mars
{"title":"The constraint tensor for null hypersurfaces","authors":"Miguel Manzano,&nbsp;Marc Mars","doi":"10.1016/j.geomphys.2024.105375","DOIUrl":"10.1016/j.geomphys.2024.105375","url":null,"abstract":"<div><div>In this work we provide a definition of the constraint tensor of a null hypersurface data which is completely explicit in the extrinsic geometry of the hypersurface. The definition is fully covariant and applies for any topology of the hypersurface. For data embedded in a spacetime, the constraint tensor coincides with the pull-back of the ambient Ricci tensor. As applications of the results, we find three geometric quantities on any transverse submanifold <em>S</em> of the data with remarkably simple gauge behaviour, and prove that the restriction of the constraint tensor to <em>S</em> takes a very simple form in terms of them. We also obtain an identity that generalizes the standard near horizon equation of isolated horizons to totally geodesic null hypersurfaces with any topology. Finally, we prove that when a null hypersurface has product topology, its extrinsic curvature can be uniquely reconstructed from the constraint tensor plus suitable initial data on a cross-section.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"208 ","pages":"Article 105375"},"PeriodicalIF":1.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dirac generating operators of split Courant algebroids 分裂库朗梯形的狄拉克生成算子
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-22 DOI: 10.1016/j.geomphys.2024.105373
Liqiang Cai , Zhuo Chen , Honglei Lang , Maosong Xiang
{"title":"Dirac generating operators of split Courant algebroids","authors":"Liqiang Cai ,&nbsp;Zhuo Chen ,&nbsp;Honglei Lang ,&nbsp;Maosong Xiang","doi":"10.1016/j.geomphys.2024.105373","DOIUrl":"10.1016/j.geomphys.2024.105373","url":null,"abstract":"<div><div>Given a vector bundle <em>A</em> over a smooth manifold <em>M</em> such that the square root <span><math><mi>L</mi></math></span> of the line bundle <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mi>top</mi></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>⊗</mo><msup><mrow><mo>∧</mo></mrow><mrow><mi>top</mi></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>M</mi></math></span> exists, the Clifford bundle associated to the standard split pseudo-Euclidean vector bundle <span><math><mo>(</mo><mi>E</mi><mo>=</mo><mi>A</mi><mo>⊕</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>,</mo><mo>〈</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>〉</mo><mo>)</mo></math></span> admits a spinor bundle <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mo>•</mo></mrow></msup><mi>A</mi><mo>⊗</mo><mi>L</mi></math></span>, whose section space consists of Berezinian half-densities of the graded manifold <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>[</mo><mn>1</mn><mo>]</mo></math></span>. Inspired by Kosmann-Schwarzbach's formula of deriving operator of split Courant algebroid (or proto-bialgebroid) structures on <span><math><mi>A</mi><mo>⊕</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, we give an explicit construction of the associate Dirac generating operator introduced by Alekseev and Xu. We prove that the square of the Dirac generating operator is an invariant of the corresponding split Courant algebroid, and also give an explicit expression of this invariant in terms of modular elements.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"208 ","pages":"Article 105373"},"PeriodicalIF":1.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Construction of Markov partitions for the geodesic flow on compact Riemann surfaces of constant negative curvature 为负曲率恒定的紧凑黎曼曲面上的大地流构建马尔可夫分区
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-22 DOI: 10.1016/j.geomphys.2024.105374
Huynh M. Hien
{"title":"Construction of Markov partitions for the geodesic flow on compact Riemann surfaces of constant negative curvature","authors":"Huynh M. Hien","doi":"10.1016/j.geomphys.2024.105374","DOIUrl":"10.1016/j.geomphys.2024.105374","url":null,"abstract":"<div><div>It is well-known that hyperbolic flows admit Markov partitions of arbitrarily small size. However, the constructions of Markov partitions for general hyperbolic flows are quite abstract and not easy to understand. To establish a more detailed understanding of Markov partitions, in this paper we consider the geodesic flow on Riemann surfaces of constant negative curvature. We provide a more complete construction of Markov partitions for this hyperbolic flow with explicit forms of rectangles and local cross sections. The local product structure is also calculated in detail.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"208 ","pages":"Article 105374"},"PeriodicalIF":1.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Abelianization of Lie algebroids and Lie groupoids 李代数和李群的无差别化
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-17 DOI: 10.1016/j.geomphys.2024.105372
Shuyu Xiao
{"title":"Abelianization of Lie algebroids and Lie groupoids","authors":"Shuyu Xiao","doi":"10.1016/j.geomphys.2024.105372","DOIUrl":"10.1016/j.geomphys.2024.105372","url":null,"abstract":"<div><div>We investigate the abelianization of a Lie algebroid and provide a necessary and sufficient condition for its existence. We also study the abelianization of groupoids and provide sufficient conditions for its existence in the smooth category and a necessary and sufficient condition for its existence in the diffeological category.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"207 ","pages":"Article 105372"},"PeriodicalIF":1.6,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation 仿射流形：多维一致 TED 方程的微分几何学
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-14 DOI: 10.1016/j.geomphys.2024.105366
W.K. Schief , U. Hertrich-Jeromin , B.G. Konopelchenko
{"title":"Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation","authors":"W.K. Schief ,&nbsp;U. Hertrich-Jeromin ,&nbsp;B.G. Konopelchenko","doi":"10.1016/j.geomphys.2024.105366","DOIUrl":"10.1016/j.geomphys.2024.105366","url":null,"abstract":"<div><div>It is shown that a canonical geometric setting of the integrable TED equation is a Kählerian tangent bundle of an affine manifold. The remarkable multi-dimensional consistency of this 4+4-dimensional dispersionless partial differential equation arises naturally in this context. In a particular 4-dimensional reduction, the affine manifolds turn out to be self-dual Einstein spaces of neutral signature governed by Plebański's first heavenly equation. In another reduction, the affine manifolds are Hessian, governed by compatible general heavenly equations. The Legendre invariance of the latter gives rise to a (dual) Hessian structure. Foliations of affine manifolds in terms of self-dual Einstein spaces are also shown to arise in connection with a natural 5-dimensional reduction.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"207 ","pages":"Article 105366"},"PeriodicalIF":1.6,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Klein-Gordon oscillators and Bergman spaces 克莱因-戈登振荡器和伯格曼空间
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-14 DOI: 10.1016/j.geomphys.2024.105368
Alexander D. Popov
{"title":"Klein-Gordon oscillators and Bergman spaces","authors":"Alexander D. Popov","doi":"10.1016/j.geomphys.2024.105368","DOIUrl":"10.1016/j.geomphys.2024.105368","url":null,"abstract":"<div><div>We consider classical and quantum dynamics of relativistic oscillator in Minkowski space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span>. It is shown that for a non-zero frequency parameter <em>ω</em> the covariant phase space of the classical Klein-Gordon oscillator is a homogeneous Kähler-Einstein manifold <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>=</mo><mrow><mi>Ad</mi></mrow><msub><mrow><mi>S</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>/</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>U</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. In the limit <span><math><mi>ω</mi><mo>→</mo><mn>0</mn></math></span>, this manifold is deformed into the covariant phase space <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> of a free relativistic particle, where <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>3</mn></mrow></msubsup><mo>∪</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mo>−</mo></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> is a two-sheeted hyperboloid in momentum space. Quantization of this model with <span><math><mi>ω</mi><mo>≠</mo><mn>0</mn></math></span> leads to the Klein-Gordon oscillator equation which we consider in the Segal-Bargmann representation. It is shown that the general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic (for antiparticles) functions on the Kähler-Einstein manifold <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span>. This relativistic model is Lorentz covariant, unitary and does not contain non-physical states.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"207 ","pages":"Article 105368"},"PeriodicalIF":1.6,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hom-actions and class equation for Hom-groups 同域群的同作用和类方程
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-13 DOI: 10.1016/j.geomphys.2024.105371
Zoheir Chebel , Hadjer Adimi , Hassane Bouremel
{"title":"Hom-actions and class equation for Hom-groups","authors":"Zoheir Chebel ,&nbsp;Hadjer Adimi ,&nbsp;Hassane Bouremel","doi":"10.1016/j.geomphys.2024.105371","DOIUrl":"10.1016/j.geomphys.2024.105371","url":null,"abstract":"<div><div>The notion of Hom-groups is defined as a generalization of a non-associative group. They can be obtained by twisting the associative operation with a compatible bijection mapping. In this article, we provide some constructions by twisting and also discuss properties related to Hom-groups. We introduce different notions of actions concerning Hom-groups. We then present a theorem for a class equation, which is proven. Following that, we illustrate some applications for <em>p</em>-Hom groups.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"207 ","pages":"Article 105371"},"PeriodicalIF":1.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Tau functions of modified CKP hierarchy 修改后的 CKP 层次结构的 Tau 功能
IF 1.6 3区 数学
Journal of Geometry and Physics Pub Date : 2024-11-13 DOI: 10.1016/j.geomphys.2024.105367
Shen Wang , Wenchuang Guan , Jipeng Cheng
{"title":"Tau functions of modified CKP hierarchy","authors":"Shen Wang ,&nbsp;Wenchuang Guan ,&nbsp;Jipeng Cheng","doi":"10.1016/j.geomphys.2024.105367","DOIUrl":"10.1016/j.geomphys.2024.105367","url":null,"abstract":"<div><div>Modified CKP (mCKP) hierarchy is an important integrable hierarchy, that is related to CKP hierarchy through Miura link. It has been proven that there exists a tau pair <span><math><mo>(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> for mCKP hierarchy. Further we find that mCKP hierarchy can be fully determined by CKP tau function and corresponding CKP eigenfunction. Based on this, we construct mCKP tau functions by CKP Darboux transformations and also present the vacuum expectation value of free bosons. As a byproduct, determinant formula for <span><math><mo>〈</mo><mn>1</mn><mo>|</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>H</mi><mo>(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>)</mo></mrow></msup><msub><mrow><mi>β</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mfrac><mrow><msubsup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mfrac><mrow><msubsup><mrow><mi>β</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>⋯</mo><msup><mrow><mi>e</mi></mrow><mrow><mfrac><mrow><msubsup><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mi>g</mi><mo>|</mo><mn>0</mn><mo>〉</mo></math></span> is also derived.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"207 ","pages":"Article 105367"},"PeriodicalIF":1.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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