Pub Date : 2025-12-17DOI: 10.1016/j.geomphys.2025.105739
Yujie Li, Nan Liu
This paper systematically studies the discrete modified Korteweg–de Vries (mKdV) equation with arbitrary-order poles based on the Riemann–Hilbert (RH) approach. Firstly, in the direct scattering problem, we present a complete analysis for the analyticity, asymptotic behaviors, and symmetries of the Jost solutions and scattering data. In particular, a detailed analysis of the discrete spectrum associated with 2N pairs arbitrary-order poles is provided. Secondly, in the inverse scattering problem, we construct a canonical matrix RH problem with residue conditions characterized at these 2N pairs of poles. By solving the RH problem, we derive the reconstruction formula for the solution of the discrete mKdV equation. Finally, in the reflectionless case, the inverse problem can be reduced to a set of linear algebraic equations, which allows us to provide an explicit parametric representation of higher-order soliton solutions.
{"title":"Riemann–Hilbert approach for discrete mKdV equation with arbitrary-order poles","authors":"Yujie Li, Nan Liu","doi":"10.1016/j.geomphys.2025.105739","DOIUrl":"10.1016/j.geomphys.2025.105739","url":null,"abstract":"<div><div>This paper systematically studies the discrete modified Korteweg–de Vries (mKdV) equation with arbitrary-order poles based on the Riemann–Hilbert (RH) approach. Firstly, in the direct scattering problem, we present a complete analysis for the analyticity, asymptotic behaviors, and symmetries of the Jost solutions and scattering data. In particular, a detailed analysis of the discrete spectrum associated with 2<em>N</em> pairs arbitrary-order poles is provided. Secondly, in the inverse scattering problem, we construct a canonical <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrix RH problem with residue conditions characterized at these 2<em>N</em> pairs of poles. By solving the RH problem, we derive the reconstruction formula for the solution of the discrete mKdV equation. Finally, in the reflectionless case, the inverse problem can be reduced to a set of linear algebraic equations, which allows us to provide an explicit parametric representation of higher-order soliton solutions.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"221 ","pages":"Article 105739"},"PeriodicalIF":1.2,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841321","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}
Pub Date : 2025-12-08DOI: 10.1016/j.geomphys.2025.105738
Manuel Ladra , Pilar Páez-Guillán
We prove an eight-term exact sequence in the homology of Lie superalgebras. We use the technique of the non-abelian tensor product to prove Schur- and Baer-type theorems for Lie superalgebras.
{"title":"The non-abelian tensor product of Lie superalgebras and Schur- and Baer-type theorems","authors":"Manuel Ladra , Pilar Páez-Guillán","doi":"10.1016/j.geomphys.2025.105738","DOIUrl":"10.1016/j.geomphys.2025.105738","url":null,"abstract":"<div><div>We prove an eight-term exact sequence in the homology of Lie superalgebras. We use the technique of the non-abelian tensor product to prove Schur- and Baer-type theorems for Lie superalgebras.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"221 ","pages":"Article 105738"},"PeriodicalIF":1.2,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747575","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}
Pub Date : 2025-12-03DOI: 10.1016/j.geomphys.2025.105723
Andrey Losev , Dmitrii Sheptunov , Xin Geng
One of the approaches to quantum gravity is to formulate it in terms of de Rham algebra, choose a triangulation of space-time, and replace differential forms by cochains (that form a finite dimensional vector space). The key issue in general relativity is the action of diffeomorphisms of space-time on fields. In this paper, we induce the action of diffeomorphisms on cochains by homotopy transfer (or, equivalently, BV integral) that leads to an action. We explicitly compute this action for the space-time, being an interval and a circle.
{"title":"On induced L∞ action of diffeomorphisms on cochains","authors":"Andrey Losev , Dmitrii Sheptunov , Xin Geng","doi":"10.1016/j.geomphys.2025.105723","DOIUrl":"10.1016/j.geomphys.2025.105723","url":null,"abstract":"<div><div>One of the approaches to quantum gravity is to formulate it in terms of de Rham algebra, choose a triangulation of space-time, and replace differential forms by cochains (that form a finite dimensional vector space). The key issue in general relativity is the action of diffeomorphisms of space-time on fields. In this paper, we induce the action of diffeomorphisms on cochains by homotopy transfer (or, equivalently, BV integral) that leads to an <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> action. We explicitly compute this action for the space-time, being an interval and a circle.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"221 ","pages":"Article 105723"},"PeriodicalIF":1.2,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145697786","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}
Pub Date : 2025-12-01DOI: 10.1016/j.geomphys.2025.105721
Ron Donagi , Nadia Ott
{"title":"Supermoduli space with Ramond punctures is not projected","authors":"Ron Donagi , Nadia Ott","doi":"10.1016/j.geomphys.2025.105721","DOIUrl":"10.1016/j.geomphys.2025.105721","url":null,"abstract":"","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"220 ","pages":"Article 105721"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693740","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}
Pub Date : 2025-12-01DOI: 10.1016/j.geomphys.2025.105730
Dongmei Zhang , Fangyang Zheng
In 1976, Milnor classified all Lie groups admitting a flat left-invariant metric. They form a special type of unimodular 2-step solvable groups. Considering Lie groups with Hermitian structure, namely, a left-invariant complex structure and a compatible left-invariant metric, in 2006, Barberis-Dotti-Fino obtained among other things full classification of all Lie groups with Hermitian structure that are Kähler and flat. In this note, we examine Lie groups with a Hermitian structure that are flat, and show that they actually must be Kähler, or equivalently speaking, a flat Hermitian Lie algebra is always Kähler. In the proofs we utilized analysis on the Hermitian geometry of 2-step solvable Lie groups developed by Freibert-Swann and by Chen and the second named author.
{"title":"Flat Hermitian Lie algebras are Kähler","authors":"Dongmei Zhang , Fangyang Zheng","doi":"10.1016/j.geomphys.2025.105730","DOIUrl":"10.1016/j.geomphys.2025.105730","url":null,"abstract":"<div><div>In 1976, Milnor classified all Lie groups admitting a flat left-invariant metric. They form a special type of unimodular 2-step solvable groups. Considering Lie groups with Hermitian structure, namely, a left-invariant complex structure and a compatible left-invariant metric, in 2006, Barberis-Dotti-Fino obtained among other things full classification of all Lie groups with Hermitian structure that are Kähler and flat. In this note, we examine Lie groups with a Hermitian structure that are flat, and show that they actually must be Kähler, or equivalently speaking, a flat Hermitian Lie algebra is always Kähler. In the proofs we utilized analysis on the Hermitian geometry of 2-step solvable Lie groups developed by Freibert-Swann and by Chen and the second named author.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"220 ","pages":"Article 105730"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693744","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}
Pub Date : 2025-12-01DOI: 10.1016/j.geomphys.2025.105722
Oleg I. Morozov
We combine the construction of the canonical conservation law and the nonlocal cosymmetry to derive a collection of nonlocal conservation laws for the two-dimensional Euler equation in vorticity form. For computational convenience and simplicity of presentation of the results we perform a complex rotation of the independent variables.
{"title":"Nonlocal conservation laws for the two-dimensional Euler equation in vorticity form","authors":"Oleg I. Morozov","doi":"10.1016/j.geomphys.2025.105722","DOIUrl":"10.1016/j.geomphys.2025.105722","url":null,"abstract":"<div><div>We combine the construction of the canonical conservation law and the nonlocal cosymmetry to derive a collection of nonlocal conservation laws for the two-dimensional Euler equation in vorticity form. For computational convenience and simplicity of presentation of the results we perform a complex rotation of the independent variables.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"220 ","pages":"Article 105722"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693743","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}
Pub Date : 2025-11-24DOI: 10.1016/j.geomphys.2025.105719
Soumalya Joardar, Atibur Rahaman
This is a continuation of the work done by the authors in [5]. In [5], an almost complex structure on finitely many points from bidirected polygon was introduced. In this paper we study a Kähler structure on finite set of points. In particular, we study the edge Laplacian of a graph twisted by the Kähler structure introduced in this paper. We also discuss a metric aspect from a twisted holomorphic Dolbeault–Dirac spectral triple and show that the points have a finite diameter with respect to the Connes' distance.
{"title":"Twisted edge Laplacians on finite graphs from a Kähler structure","authors":"Soumalya Joardar, Atibur Rahaman","doi":"10.1016/j.geomphys.2025.105719","DOIUrl":"10.1016/j.geomphys.2025.105719","url":null,"abstract":"<div><div>This is a continuation of the work done by the authors in <span><span>[5]</span></span>. In <span><span>[5]</span></span>, an almost complex structure on finitely many points from bidirected polygon was introduced. In this paper we study a Kähler structure on finite set of points. In particular, we study the edge Laplacian of a graph twisted by the Kähler structure introduced in this paper. We also discuss a metric aspect from a twisted holomorphic Dolbeault–Dirac spectral triple and show that the points have a finite diameter with respect to the Connes' distance.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"220 ","pages":"Article 105719"},"PeriodicalIF":1.2,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624554","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}
Pub Date : 2025-11-24DOI: 10.1016/j.geomphys.2025.105720
G.B. Shabat , V.V. Sokolov , A.V. Tsiganov
We consider Novikov equations for commutative ring generated by differential operators of orders 3,4,5. We present an explicit Hamiltonian form of these equations. Using the method of compatible Poisson brackets, we find a separation of variables on a hyperelliptic curve of genus 2 for the Novikov equations.
{"title":"Novikov equations for commuting differential operators of orders 3,4,5","authors":"G.B. Shabat , V.V. Sokolov , A.V. Tsiganov","doi":"10.1016/j.geomphys.2025.105720","DOIUrl":"10.1016/j.geomphys.2025.105720","url":null,"abstract":"<div><div>We consider Novikov equations for commutative ring generated by differential operators of orders 3,4,5. We present an explicit Hamiltonian form of these equations. Using the method of compatible Poisson brackets, we find a separation of variables on a hyperelliptic curve of genus 2 for the Novikov equations.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"220 ","pages":"Article 105720"},"PeriodicalIF":1.2,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624551","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}
Pub Date : 2025-11-21DOI: 10.1016/j.geomphys.2025.105714
Khaled Basdouri , Ghaith Chaabane
Recent research has extensively explored Bihom-structures. In this paper, we introduce the notion of double Bihom–Lie superbialgebras and develop a corresponding cohomology theory, defined as the total cohomology of a double complex constructed from a Bihom–Lie superalgebra and its dual. We demonstrate that the second cohomology group classifies formal deformations of Bihom–Lie superbialgebras. Furthermore, we provide explicit computations and examples in low-dimensional cases to illustrate these results.
{"title":"Cohomology of Bihom-Lie superbialgebras","authors":"Khaled Basdouri , Ghaith Chaabane","doi":"10.1016/j.geomphys.2025.105714","DOIUrl":"10.1016/j.geomphys.2025.105714","url":null,"abstract":"<div><div>Recent research has extensively explored Bihom-structures. In this paper, we introduce the notion of double Bihom–Lie superbialgebras and develop a corresponding cohomology theory, defined as the total cohomology of a double complex constructed from a Bihom–Lie superalgebra and its dual. We demonstrate that the second cohomology group classifies formal deformations of Bihom–Lie superbialgebras. Furthermore, we provide explicit computations and examples in low-dimensional cases to illustrate these results.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"221 ","pages":"Article 105714"},"PeriodicalIF":1.2,"publicationDate":"2025-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145697785","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}
Pub Date : 2025-11-20DOI: 10.1016/j.geomphys.2025.105717
Abraham Bobadilla Osses , Mauricio Godoy Molina
In this paper, we study the mechanical system associated with rolling a Lorentzian manifold of dimension on flat Lorentzian space , without slipping or twisting. Using previous results, it is known that there exists a distribution of rank defined on the configuration space of the rolling system, encoding the no-slip and no-twist conditions. Our objective is to study the problem of complete controllability of the control system associated with . The key lies in examining the holonomy group of the distribution and, following the approach of [7], establishing that the rolling problem is completely controllable if and only if the holonomy group of equals .
{"title":"Controllability of the rolling system of a Lorentzian manifold on Rn,1","authors":"Abraham Bobadilla Osses , Mauricio Godoy Molina","doi":"10.1016/j.geomphys.2025.105717","DOIUrl":"10.1016/j.geomphys.2025.105717","url":null,"abstract":"<div><div>In this paper, we study the mechanical system associated with rolling a Lorentzian manifold <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> of dimension <span><math><mi>n</mi><mo>+</mo><mn>1</mn><mo>≥</mo><mn>2</mn></math></span> on flat Lorentzian space <span><math><mover><mrow><mi>M</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>1</mn></mrow></msup></math></span>, without slipping or twisting. Using previous results, it is known that there exists a distribution <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> of rank <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> defined on the configuration space <span><math><mi>Q</mi><mo>(</mo><mi>M</mi><mo>,</mo><mover><mrow><mi>M</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> of the rolling system, encoding the no-slip and no-twist conditions. Our objective is to study the problem of complete controllability of the control system associated with <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>. The key lies in examining the holonomy group of the distribution <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> and, following the approach of <span><span>[7]</span></span>, establishing that the rolling problem is completely controllable if and only if the holonomy group of <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> equals <span><math><mi>S</mi><msub><mrow><mi>O</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"220 ","pages":"Article 105717"},"PeriodicalIF":1.2,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624553","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}
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