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From Auto-Bäcklund Transformations to Auto-Bäcklund Transformations, and Torqued ABS Equations 从Auto-Bäcklund变换到Auto-Bäcklund变换，以及扭力ABS方程
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-09-23 DOI: 10.1007/s11040-021-09406-1
Dan-da Zhang, Da-jun Zhang, Peter H. van der Kamp
{"title":"From Auto-Bäcklund Transformations to Auto-Bäcklund Transformations, and Torqued ABS Equations","authors":"Dan-da Zhang,&nbsp;Da-jun Zhang,&nbsp;Peter H. van der Kamp","doi":"10.1007/s11040-021-09406-1","DOIUrl":"10.1007/s11040-021-09406-1","url":null,"abstract":"<div><p>We provide a method which from a given auto-Bäcklund transformation (auto-BT) produces another auto-BT for a different equation. We apply the method to the natural auto-BTs for the ABS quad equations, which gives rise to torqued versions of ABS equations and explains the origin of each auto-BT listed in Atkinson (J. Phys. A: Math. Theor. <b>41</b>(8pp), 135202, 2008). The method is also applied to non-natural auto-BTs for ABS equations, which yields 3D consistent cubes which have not been found in Boll (J. Nonl. Math. Phys. <b>18</b>, 337–365, 2011), and to a multi-quadratic ABS* equation giving rise to a multi-quartic equation.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4919320","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: 4
Solitons for the Modified Camassa-Holm Equation and their Interactions Via Dressing Method 修正Camassa-Holm方程的孤子及其通过修饰法的相互作用
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-09-21 DOI: 10.1007/s11040-021-09395-1
Hui Mao, Yonghui Kuang
{"title":"Solitons for the Modified Camassa-Holm Equation and their Interactions Via Dressing Method","authors":"Hui Mao,&nbsp;Yonghui Kuang","doi":"10.1007/s11040-021-09395-1","DOIUrl":"10.1007/s11040-021-09395-1","url":null,"abstract":"<div><p>In this paper, we develop the dressing method to study the modified Camassa-Holm equation with the help of reciprocal transformation and the associated modified Camassa- Holm equation. Based on this method, some different soliton solutions, in particular dark solitons to the modified Camassa-Holm equation are presented and their interactions are investigated.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09395-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5142250","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: 2
New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree Cayley树上不可数自旋值集模型Gibbs测度唯一性的新条件
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-09-20 DOI: 10.1007/s11040-021-09404-3
F. H. Haydarov
{"title":"New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree","authors":"F. H. Haydarov","doi":"10.1007/s11040-021-09404-3","DOIUrl":"10.1007/s11040-021-09404-3","url":null,"abstract":"<div><p>In this paper we consider a model with nearest-neighbor interactions with spin space [0, 1] on Cayley trees of order <i>k</i> ⩾ 2. In Yu et al. (2013), a sufficient condition of uniqueness for the splitting Gibbs measure of the model is given. We investigate the sufficient condition of uniqueness and obtain better estimates.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09404-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4809844","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: 1
Density of States and Lifshitz Tails for Discrete 1D Random Dirac Operators 离散1D随机Dirac算子的态密度和Lifshitz尾
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-09-14 DOI: 10.1007/s11040-021-09403-4
Roberto A. Prado, César R. de Oliveira, Edmundo C. de Oliveira
{"title":"Density of States and Lifshitz Tails for Discrete 1D Random Dirac Operators","authors":"Roberto A. Prado,&nbsp;César R. de Oliveira,&nbsp;Edmundo C. de Oliveira","doi":"10.1007/s11040-021-09403-4","DOIUrl":"10.1007/s11040-021-09403-4","url":null,"abstract":"<div><p>We study the density of states and Lifshitz tails for a family of random Dirac operators on the one-dimensional lattice <span>(mathbb {Z})</span>. These operators consist of the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by two different scalar potentials, which are sequences of independent and identically distributed random variables according to a Borel probability measure of compact support in <span>(mathbb {R})</span>. The existence of the density of state measure for these Dirac operators is obtained through two approaches by finite-volume quantities. By using one of these approaches, we show that the distribution function of the density of states decays exponentially for energies near the spectral band edges, i.e., we establish Lifshitz tails for these operators. Lifshitz tails are established first for Dirac operators restricted to appropriate subspaces of energies and, using this, extended to the full operators, including the occurrence of internal tails in the case of spectral gap.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4590762","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: 4
Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides 右手边为二次多项式的两个耦合一阶ode系统的解
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-08-17 DOI: 10.1007/s11040-021-09400-7
Francesco Calogero, Farrin Payandeh
{"title":"Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides","authors":"Francesco Calogero,&nbsp;Farrin Payandeh","doi":"10.1007/s11040-021-09400-7","DOIUrl":"10.1007/s11040-021-09400-7","url":null,"abstract":"<div><p>The <i>explicit</i> solution <span>(x_{n}left (tright ) ,)</span> <i>n</i> = 1,2, of the <i>initial-values</i> problem is exhibited of a <i>subclass</i> of the <i>autonomous</i> system of 2 coupled <i>first-order</i> ODEs with <i>second-degree</i> polynomial right-hand sides, hence featuring 12 <i>a priori arbitrary</i> (time-independent) coefficients: \u0000</p><div><div><span>$$ dot{x}_{n}=c_{n1}left( x_{1}right)^{2}+c_{n2}x_{1}x_{2}+c_{n3}left( x_{2}right)^{2}+c_{n4}x_{1}+c_{n5}x_{2}+c_{n6}~,~~~n=1,2~. $$</span></div></div><p> The solution is <i>explicitly</i> provided if the 12 coefficients <i>c</i><sub><i>n</i><i>j</i></sub> (<i>n</i> = 1,2; <i>j</i> = 1,2,3,4,5,6) are expressed by <i>explicitly</i> provided formulas in terms of 10 <i>a priori arbitrary</i> parameters; the <i>inverse</i> problem to express these 10 parameters in terms of the 12 coefficients <i>c</i><sub><i>n</i><i>j</i></sub> is also <i>explicitly</i> solved, but it is found to imply—as it were, <i>a posteriori</i>—that the 12 coefficients <i>c</i><sub><i>n</i><i>j</i></sub> must then satisfy 4 <i>algebraic constraints</i>, which are <i>explicitly</i> exhibited. Special subcases are also identified the <i>general</i> solutions of which are <i>completely periodic</i> with a period independent of the initial data (“isochrony”), or are characterized by additional restrictions on the coefficients <i>c</i><sub><i>n</i><i>j</i></sub> which identify particularly interesting models.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09400-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4668589","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: 3
Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes 渐近ad时空上具有任意边界条件的标量场的基本解和Hadamard态
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-08-12 DOI: 10.1007/s11040-021-09402-5
Claudio Dappiaggi, Alessio Marta
{"title":"Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes","authors":"Claudio Dappiaggi,&nbsp;Alessio Marta","doi":"10.1007/s11040-021-09402-5","DOIUrl":"10.1007/s11040-021-09402-5","url":null,"abstract":"<div><p>We consider the Klein-Gordon operator on an <i>n</i>-dimensional asymptotically anti-de Sitter spacetime (<i>M</i>,<i>g</i>) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on <i>∂</i><i>M</i> of order up to 2. Using techniques from <i>b</i>-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09402-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4484318","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: 9
Bilinear Equation and Additional Symmetries for an Extension of the Kadomtsev–Petviashvili Hierarchy Kadomtsev-Petviashvili层次扩展的双线性方程和附加对称性
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-08-06 DOI: 10.1007/s11040-021-09401-6
Jiaping Lu, Chao-Zhong Wu
{"title":"Bilinear Equation and Additional Symmetries for an Extension of the Kadomtsev–Petviashvili Hierarchy","authors":"Jiaping Lu,&nbsp;Chao-Zhong Wu","doi":"10.1007/s11040-021-09401-6","DOIUrl":"10.1007/s11040-021-09401-6","url":null,"abstract":"<div><p>An extension of the Kadomtsev–Petviashvili (KP) hierarchy defined via scalar pseudo-differential operators was studied in Szablikowski and Blaszak (J. Math. Phys. <b>49</b>(8), 082701, 20, 2008) and Wu and Zhou (J. Geom. Phys. <b>106</b>, 327–341, 2016). In this paper, we represent the extended KP hierarchy into the form of bilinear equation of (adjoint) Baker–Akhiezer functions, and construct its additional symmetries. As a byproduct, we derive the Virasoro symmetries for the constrained KP hierarchies.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09401-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4238213","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: 7
Kahan Discretizations of Skew-Symmetric Lotka-Volterra Systems and Poisson Maps 斜对称Lotka-Volterra系统和泊松映射的Kahan离散化
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-07-30 DOI: 10.1007/s11040-021-09399-x
C. A. Evripidou, P. Kassotakis, P. Vanhaecke
{"title":"Kahan Discretizations of Skew-Symmetric Lotka-Volterra Systems and Poisson Maps","authors":"C. A. Evripidou,&nbsp;P. Kassotakis,&nbsp;P. Vanhaecke","doi":"10.1007/s11040-021-09399-x","DOIUrl":"10.1007/s11040-021-09399-x","url":null,"abstract":"<div><p>The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph Γ, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic Poisson structure of the Lotka-Volterra system, we say that the graph Γ has the Kahan-Poisson property. We show that if Γ is connected, it has the Kahan-Poisson property if and only if it is a cloning of a graph with vertices <span>(1,2,dots ,n)</span>, with an arc <i>i</i> → <i>j</i> precisely when <i>i</i> &lt; <i>j</i>, and with all arcs having the same value. We also prove a similar result for augmented graphs, which correspond with deformed Lotka-Volterra systems and show that the obtained Lotka-Volterra systems and their Kahan discretizations are superintegrable as well as Liouville integrable.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09399-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5147330","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: 1
Discrete mKdV Equation via Darboux Transformation 基于达布变换的离散mKdV方程
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-07-14 DOI: 10.1007/s11040-021-09398-y
Joseph Cho, Wayne Rossman, Tomoya Seno
{"title":"Discrete mKdV Equation via Darboux Transformation","authors":"Joseph Cho,&nbsp;Wayne Rossman,&nbsp;Tomoya Seno","doi":"10.1007/s11040-021-09398-y","DOIUrl":"10.1007/s11040-021-09398-y","url":null,"abstract":"<div><p>We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09398-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4000475","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: 2
Asymptotic Behavior of Density in the Boundary-Driven Exclusion Process on the Sierpinski Gasket Sierpinski垫片边界驱动不相容过程中密度的渐近行为
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2021-07-08 DOI: 10.1007/s11040-021-09392-4
Joe P. Chen, Patrícia Gonçalves
{"title":"Asymptotic Behavior of Density in the Boundary-Driven Exclusion Process on the Sierpinski Gasket","authors":"Joe P. Chen,&nbsp;Patrícia Gonçalves","doi":"10.1007/s11040-021-09392-4","DOIUrl":"10.1007/s11040-021-09392-4","url":null,"abstract":"<div><p>We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09392-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4339805","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: 2
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