Pub Date : 2025-02-01DOI: 10.1016/S0034-4877(25)00010-2
Shahid Ahmad Wani, Mohra Zayed, Hassan Ali, Subuhi Khan
This paper presents a novel framework for introducing generalized 1-parameter 3-variable Hermite–Frobenius–Euler polynomials. These polynomials are characterized by generating functions, series definitions, and summation formulae, elucidating their fundamental properties. Moreover, the study establishes recurrence relations, shift operators, and various differential equations, including differential, integro-differential, and partial differential equations, utilizing a factorization method.
{"title":"Families of Differential Equations Associated with The Generalized 1-Parameter 3-Variable Hermite–Frobenius–Euler Polynomials","authors":"Shahid Ahmad Wani, Mohra Zayed, Hassan Ali, Subuhi Khan","doi":"10.1016/S0034-4877(25)00010-2","DOIUrl":"10.1016/S0034-4877(25)00010-2","url":null,"abstract":"<div><div>This paper presents a novel framework for introducing generalized 1-parameter 3-variable Hermite–Frobenius–Euler polynomials. These polynomials are characterized by generating functions, series definitions, and summation formulae, elucidating their fundamental properties. Moreover, the study establishes recurrence relations, shift operators, and various differential equations, including differential, integro-differential, and partial differential equations, utilizing a factorization method.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"95 1","pages":"Pages 53-70"},"PeriodicalIF":1.0,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143680613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/S0034-4877(25)00011-4
Ahmet Kazan, Mustafa Altin
In this paper, for a soliton surface Ω = Ω(u, v) associated with the Betchov–Da Rios equation, we obtain the derivative formulae of an extended Darboux frame field of a unit speed u-parameter curve Ω = Ω(u, v) for all v. Also, we get the geometric invariants k and h of the soliton surface Ω = Ω(u, v) and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of Ω. We give some important geometric characterizations such as flatness, minimality and semi-umbilicaly with the aid of these invariants. Additionally, we study the curvature ellipse of the Betchov–Da Rios soliton surface and Wintgen ideal (superconformal) Betchov–Da Rios soliton surface with respect to an extended Darboux frame field. Finally, we construct an application for the Betchov–Da Rios soliton surface with the aid of an extended Darboux frame field.
{"title":"A Geometric Application of Soliton Surfaces Associated with The Betchov–Da Rios Equation Using an Extended Darboux Frame Field in E4","authors":"Ahmet Kazan, Mustafa Altin","doi":"10.1016/S0034-4877(25)00011-4","DOIUrl":"10.1016/S0034-4877(25)00011-4","url":null,"abstract":"<div><div>In this paper, for a soliton surface Ω = Ω(<em>u, v</em>) associated with the Betchov–Da Rios equation, we obtain the derivative formulae of an extended Darboux frame field of a unit speed <em>u</em>-parameter curve Ω = Ω(<em>u, v</em>) for all <em>v.</em> Also, we get the geometric invariants <em>k</em> and <em>h</em> of the soliton surface Ω = Ω(<em>u, v</em>) and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of Ω. We give some important geometric characterizations such as flatness, minimality and semi-umbilicaly with the aid of these invariants. Additionally, we study the curvature ellipse of the Betchov–Da Rios soliton surface and Wintgen ideal (superconformal) Betchov–Da Rios soliton surface with respect to an extended Darboux frame field. Finally, we construct an application for the Betchov–Da Rios soliton surface with the aid of an extended Darboux frame field.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"95 1","pages":"Pages 71-92"},"PeriodicalIF":1.0,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143680614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/S0034-4877(25)00003-5
Xiangyu Chen, Qiang Lei
In this paper, we introduce two measures for the resource theory of imaginarity. One is induced by α-z-Rényi relative entropy and the other, defined for positive-definite density matrices, is induced by Tsallis relative operator entropy. The relationships between different imaginarity measures and their properties are also discussed.
{"title":"Imaginarity Measures Induced by Relative Entropy","authors":"Xiangyu Chen, Qiang Lei","doi":"10.1016/S0034-4877(25)00003-5","DOIUrl":"10.1016/S0034-4877(25)00003-5","url":null,"abstract":"<div><div>In this paper, we introduce two measures for the resource theory of imaginarity. One is induced by α-<em>z</em>-Rényi relative entropy and the other, defined for positive-definite density matrices, is induced by Tsallis relative operator entropy. The relationships between different imaginarity measures and their properties are also discussed.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"95 1","pages":"Pages 1-10"},"PeriodicalIF":1.0,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143680672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-01DOI: 10.1016/S0034-4877(24)00082-X
Muhayyo A. Rasulova
We consider a (q + 1)-state Potts model with nonzero external field on the Cayley tree. We describe all ground states for this model with a translation-invariant external field and all two-periodic ground states with a periodic external field.
{"title":"Ground states for the potts model with an external field","authors":"Muhayyo A. Rasulova","doi":"10.1016/S0034-4877(24)00082-X","DOIUrl":"10.1016/S0034-4877(24)00082-X","url":null,"abstract":"<div><div>We consider a <em>(q +</em> 1)-state Potts model with nonzero external field on the Cayley tree. We describe all ground states for this model with a translation-invariant external field and all two-periodic ground states with a periodic external field.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 3","pages":"Pages 325-334"},"PeriodicalIF":1.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143317063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-01DOI: 10.1016/S0034-4877(24)00080-6
Guang-Hao Zhang, Fang-Cheng Fan
Beginning with a more generalized discrete 2 × 2 matrix spectral problem and applying the Tu scheme, a mixed integrable lattice hierarchy based on the negative and positive lattice hierarchies is constructed, it includes the well-known relativistic Toda lattice hierarchy and can reduce to other new integrable lattice hierarchies. For the first nontrivial lattice equation in the mixed hierarchy, the corresponding infinite number of conservation laws and N-fold Darboux transformation are established on the base of its Lax pair. As an application of the obtained Darboux transformation, we obtain the discrete N-fold explicit solutions in determinant form, from which we get one-and two-soliton solutions with proper parameters and their dynamical properties and evolutions are illustrated graphically. Some interesting soliton structures are presented, such as kink and bell-shaped two-soliton, bell and anti-bell shaped two-soliton and anti-bell shaped two-soliton and so on. What is more, we observe that these solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions, which means they are much stable during the propagation. These results and properties given in this paper may help us better understand nonlinear lattice dynamics.
{"title":"A mixed integrable lattice hierarchy associated with the relativistic toda lattice: conservation laws, N-fold Darboux transformation and soliton solutions","authors":"Guang-Hao Zhang, Fang-Cheng Fan","doi":"10.1016/S0034-4877(24)00080-6","DOIUrl":"10.1016/S0034-4877(24)00080-6","url":null,"abstract":"<div><div>Beginning with a more generalized discrete 2 × 2 matrix spectral problem and applying the Tu scheme, a mixed integrable lattice hierarchy based on the negative and positive lattice hierarchies is constructed, it includes the well-known relativistic Toda lattice hierarchy and can reduce to other new integrable lattice hierarchies. For the first nontrivial lattice equation in the mixed hierarchy, the corresponding infinite number of conservation laws and <em>N</em>-fold Darboux transformation are established on the base of its Lax pair. As an application of the obtained Darboux transformation, we obtain the discrete <em>N</em>-fold explicit solutions in determinant form, from which we get one-and two-soliton solutions with proper parameters and their dynamical properties and evolutions are illustrated graphically. Some interesting soliton structures are presented, such as kink and bell-shaped two-soliton, bell and anti-bell shaped two-soliton and anti-bell shaped two-soliton and so on. What is more, we observe that these solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions, which means they are much stable during the propagation. These results and properties given in this paper may help us better understand nonlinear lattice dynamics.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 3","pages":"Pages 279-304"},"PeriodicalIF":1.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143317283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The initial boundary-value problem for a plate equation with nonlocal damping is considered. The local existence of solution is proved by the monotone operator theory with locally Lipschitz perturbation. By using the potential well theory, we get the global existence of solution. By Nakao's inequality, the decay estimate of polynomial type is obtained. We provide also the sufficient conditions of finite time blow-up of weak solutions with suitable conditions on the initial data by an ordinary differential inequality for an appropriately chosen functional.
{"title":"The existence, asymptotic behaviour and blow-up of solution of a plate equation with nonlinear averaged damping","authors":"Hongwei Zhang, Ling Liu, Hongyun Yue, Donghao Li, Khaled Zennir","doi":"10.1016/S0034-4877(24)00081-8","DOIUrl":"10.1016/S0034-4877(24)00081-8","url":null,"abstract":"<div><div>The initial boundary-value problem for a plate equation with nonlocal damping is considered. The local existence of solution is proved by the monotone operator theory with locally Lipschitz perturbation. By using the potential well theory, we get the global existence of solution. By Nakao's inequality, the decay estimate of polynomial type is obtained. We provide also the sufficient conditions of finite time blow-up of weak solutions with suitable conditions on the initial data by an ordinary differential inequality for an appropriately chosen functional.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 3","pages":"Pages 305-323"},"PeriodicalIF":1.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143317066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-01DOI: 10.1016/S0034-4877(24)00087-9
Wei Feng, Songlin Zhao
This paper provides extensions of the methods of Lie symmetry group and nonlinear self–adjointness to fractional differential equations involving mixed derivatives of Riemann–Liouville time-fractional derivative and first-order partial derivative. We present explicitly the general prolongation formulae expressing the action of Lie group on the mixed fractional derivatives and the expressions of conserved vectors in conservation laws. Moreover, the obtained results are used to investigate the symmetry groups and conservation laws of time-fractional Fokas–Lenells equation, whose exact solution and nontrivial conservation law are thereby constructed.
{"title":"On invariant analysis and conservation law for fractional differential equations with mixed fractional derivative: Time-fractional Fokas–Lenells equation","authors":"Wei Feng, Songlin Zhao","doi":"10.1016/S0034-4877(24)00087-9","DOIUrl":"10.1016/S0034-4877(24)00087-9","url":null,"abstract":"<div><div>This paper provides extensions of the methods of Lie symmetry group and nonlinear self–adjointness to fractional differential equations involving mixed derivatives of Riemann–Liouville time-fractional derivative and first-order partial derivative. We present explicitly the general prolongation formulae expressing the action of Lie group on the mixed fractional derivatives and the expressions of conserved vectors in conservation laws. Moreover, the obtained results are used to investigate the symmetry groups and conservation laws of time-fractional Fokas–Lenells equation, whose exact solution and nontrivial conservation law are thereby constructed.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 3","pages":"Pages 405-420"},"PeriodicalIF":1.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143317096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-01DOI: 10.1016/S0034-4877(24)00085-5
Qijie Cao, Peng Zhao
Based on the Riemann–Hilbert method, the Riemann theta function representations for algebro-geometric solutions of the Hirota equation are derived. It is shown that the Baker–Akhiezer function of the Hirota equation can be described by solvable matrix Riemann–Hilbert problems on complex plane. The procedure avoids the use of Dubrovin's equations and Jacobi inverse problem.
{"title":"Algebro-geometric integration of the Hirota equation and the Riemann–Hilbert problem","authors":"Qijie Cao, Peng Zhao","doi":"10.1016/S0034-4877(24)00085-5","DOIUrl":"10.1016/S0034-4877(24)00085-5","url":null,"abstract":"<div><div>Based on the Riemann–Hilbert method, the Riemann theta function representations for algebro-geometric solutions of the Hirota equation are derived. It is shown that the Baker–Akhiezer function of the Hirota equation can be described by solvable matrix Riemann–Hilbert problems on complex plane. The procedure avoids the use of Dubrovin's equations and Jacobi inverse problem.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 3","pages":"Pages 365-394"},"PeriodicalIF":1.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143356489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-01DOI: 10.1016/S0034-4877(24)00086-7
Crina-Daniela Neacşu
The aim of this note is to prove that index of the identity map on a quaternion projective space of any dimension is zero. As an immediate consequence, it is established that any quaternion projective space is stable.
{"title":"On the stability of the quaternion projective space","authors":"Crina-Daniela Neacşu","doi":"10.1016/S0034-4877(24)00086-7","DOIUrl":"10.1016/S0034-4877(24)00086-7","url":null,"abstract":"<div><div>The aim of this note is to prove that index of the identity map on a quaternion projective space of any dimension is zero. As an immediate consequence, it is established that any quaternion projective space is stable.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 3","pages":"Pages 395-404"},"PeriodicalIF":1.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143317095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-01DOI: 10.1016/S0034-4877(24)00083-1
Subuhi Khan, Mahammad Lal Mia, Mahvish Ali
In this article, the Bessel and Tricomi functions are combined with Appell polynomials to introduce the families of Appell–Bessel and Appell–Tricomi functions. The 2-variable 2-parameter Hermite–Bessel and Hermite–Tricomi functions are considered as members of these families, and framed within the representation of the Lie algebra T3. Consequently, the implicit summation formulae for these functions are derived. Certain examples are also considered. The article concludes with the derivation of a relation involving the 2-variable 2-parameter Hermite–Tricomi functions by following the Weisner's approach.
{"title":"Lie algebra representation and hybrid families related to Hermite polynomials","authors":"Subuhi Khan, Mahammad Lal Mia, Mahvish Ali","doi":"10.1016/S0034-4877(24)00083-1","DOIUrl":"10.1016/S0034-4877(24)00083-1","url":null,"abstract":"<div><div>In this article, the Bessel and Tricomi functions are combined with Appell polynomials to introduce the families of Appell–Bessel and Appell–Tricomi functions. The 2-variable 2-parameter Hermite–Bessel and Hermite–Tricomi functions are considered as members of these families, and framed within the representation of the Lie algebra T3. Consequently, the implicit summation formulae for these functions are derived. Certain examples are also considered. The article concludes with the derivation of a relation involving the 2-variable 2-parameter Hermite–Tricomi functions by following the Weisner's approach.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 3","pages":"Pages 335-352"},"PeriodicalIF":1.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143317065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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