Russian Journal of Mathematical Physics最新文献

{"title":"Local Dynamics of Two-Component Parabolic Systems of Schröedinger Type","authors":"S. A. Kashchenko","doi":"10.1134/S1061920821040087","DOIUrl":"10.1134/S1061920821040087","url":null,"abstract":"<p> The local dynamics of a class of two-component nonlinear systems of parabolic equations is considered; this class is important for applications. Under sufficiently natural conditions on the coefficients of the linearized equation, the case of infinite dimension is realized, which is critical in the problem of stationary stability. An algorithm of normalization is proposed, i.e., a reduction to an infinite system of ordinary differential equations for slowly varying amplitudes. The situations are highlighted in which the corresponding systems can be compactly written in the form of boundary value problems with special nonlinearities. These boundary value problems play the role of normal forms for the original parabolic systems. Scalar complex parabolic equations of Schrödinger type are considered as important applications. The role of resonance relations when constructing nonlinear functions entering normal forms are revealed. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 4","pages":"501 - 513"},"PeriodicalIF":1.4,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4244904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}

{"title":"Pólya–Carlson Dichotomy for Dynamical Zeta Functions and a Twisted Burnside–Frobenius Theorem","authors":"A. Fel’shtyn,&nbsp;E. Troitsky","doi":"10.1134/S1061920821040051","DOIUrl":"10.1134/S1061920821040051","url":null,"abstract":"<p> For the unitary dual mapping of an automorphism of a torsion-free, finite rank nilpotent group, we prove the Pólya–Carlson dichotomy between rationality and the natural boundary for the analytic behavior of its Artin–Mazur dynamical zeta function. We also establish Gauss congruences for the Reidemeister numbers of the iterations of endomorphisms of groups in this class. Our method is the twisted Burnside–Frobenius theorem proven in the paper for automorphisms of this class of groups, and a calculation of the Reidemeister numbers via a product formula and profinite completions. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 4","pages":"455 - 463"},"PeriodicalIF":1.4,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4243986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}

{"title":"On the Correspondence between the Variational Principles in the Eulerian and Lagrangian Descriptions","authors":"A. V. Aksenov,&nbsp;K. P. Druzhkov","doi":"10.1134/S1061920821040014","DOIUrl":"10.1134/S1061920821040014","url":null,"abstract":"<p> The relationship between the variational principles for equations of continuum mechanics in Eulerian and Lagrangian descriptions is considered. It is shown that, for a system of differential equations in Eulerian variables, the corresponding Lagrangian description is related to introducing nonlocal variables. The connection between the descriptions is obtained in terms of differential coverings. The relation between the variational principles of a system of equations and its symplectic structures is discussed. It is shown that, if a system of equations in Lagrangian variables can be derived from a variational principle, then there is no corresponding variational principle in the Eulerian variables. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 4","pages":"411 - 415"},"PeriodicalIF":1.4,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4241472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}

{"title":"Complete Integrability of a New Class of Hamiltonian Hydrodynamic Type Systems","authors":"Z. V. Makridin,&nbsp;M. V. Pavlov","doi":"10.1134/S1061920821040099","DOIUrl":"10.1134/S1061920821040099","url":null,"abstract":"<p> In this paper, we consider a new class of Hamiltonian hydrodynamic type systems whose conservation laws are polynomial with respect to one of the field variables. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 4","pages":"514 - 523"},"PeriodicalIF":1.4,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4241957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}

{"title":"Continuity Criteria for Locally Bounded Automorphisms of Central Extensions of Perfect Lie Groups","authors":"A. I. Shtern","doi":"10.1134/S1061920821040117","DOIUrl":"10.1134/S1061920821040117","url":null,"abstract":"<p> We prove that every locally bounded automorphism of a linear connected Lie central extension of a connected perfect Lie group is continuous if and only if it is continuous on the center. We also prove that, if <span>(Z)</span> is a connected Abelian group without nontrivial compact subgroups, <span>(H)</span> is a connected perfect Lie group and the short sequence of Lie groups <span>({e}to Zto Gto Hto{e})</span> is exact, then every locally bounded automorphism of <span>(G)</span> is continuous if and only if it is continuous on the center of <span>(G)</span>. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 4","pages":"543 - 544"},"PeriodicalIF":1.4,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4241960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}

{"title":"The Regularized Free Fall I. Index Computations","authors":"U. Frauenfelder,&nbsp;J. Weber","doi":"10.1134/S1061920821040063","DOIUrl":"10.1134/S1061920821040063","url":null,"abstract":"<p> The main results are, firstly, a generalization of the Conley–Zehnder index from ODEs to the delay equation at hand and, secondly, the equality of the Morse index and the clockwise normalized Conley–Zehnder index <span>( mu^{rm CZ} )</span>. We consider the nonlocal Lagrangian action functional <span>( {mathcal{B}} )</span> discovered by Barutello, Ortega, and Verzini [7] with which they obtained a new regularization of the Kepler problem. Critical points of this functional are regularized periodic solutions <span>(x)</span> of the Kepler problem. In this article, we look at <i>period 1 only</i> and at dimension one (gravitational free fall). Via a nonlocal Legendre transform regularized periodic Kepler orbits <span>(x)</span> can be interpreted as periodic solutions <span>((x,y))</span> of a Hamiltonian delay equation. In particular, regularized <span>(1)</span>-periodic solutions of the free fall are represented variationally in two ways: as critical points <span>(x)</span> of a nonlocal Lagrangian action functional and as critical points <span>((x,y))</span> of a nonlocal Hamiltonian action functional. As critical points of the Lagrangian action, the <span>(1)</span>-periodic solutions have a finite Morse index which we compute first. As critical points of the Hamiltonian action <span>( {mathcal{A}} _ {mathcal{H}} )</span>, one encounters the obstacle, due to nonlocality, that the <span>(1)</span>-periodic solutions are not generated any more by a flow on the phase space manifold. Hence, the usual definition of the Conley–Zehnder index as the intersection number with a Maslov cycle is not available. In the local case, Hofer, Wysocki, and Zehnder [10] gave an alternative definition of the Conley–Zehnder index by assigning a winding number to each eigenvalue of the Hessian of <span>( {mathcal{A}} _ {mathcal{H}} )</span> at critical points. In this article, we show how to generalize the Conley–Zehnder index to the nonlocal case at hand. On one side, we discover how properties from the local case generalize to this delay equation, and on the other side, we see a new phenomenon arising. In contrast to the local case, the winding number is no longer monotone as a function of the eigenvalues. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 4","pages":"464 - 487"},"PeriodicalIF":1.4,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4242384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}

{"title":"Properties of Suns in the Spaces (L^1) and (C(Q))","authors":"I. G. Tsar’kov","doi":"10.1134/S1061920821030122","DOIUrl":"10.1134/S1061920821030122","url":null,"abstract":"<p> Properties of suns in the spaces <span>(L^1)</span> and <span>(C(Q))</span> are studied. It is shown that every boundedly compact sun in <span>(L^1)</span> is convex and every boundedly weakly compact sun in <span>(C(Q))</span> is monotone path-connected. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 3","pages":"398 - 405"},"PeriodicalIF":1.4,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4561169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}

{"title":"Some Identities on Truncated Polynomials Associated with Degenerate Bell Polynomials","authors":"T. Kim,&nbsp;D. S. Kim","doi":"10.1134/S1061920821030079","DOIUrl":"10.1134/S1061920821030079","url":null,"abstract":"<p> The aim of this paper is to introduce truncated degenerate Bell polynomials and numbers and to investigate some of their properties. In more detail, we obtain explicit expressions, identities involving other special polynomials, integral representations, a Dobinski-like formula and expressions of the generating function in terms of differential operators and the linear incomplete gamma function. In addition, we introduce truncated degenerate modified Bell polynomials and numbers and obtain similar results for those polynomials. As an application of our results, we show that the truncated degenerate Bell numbers can be expressed as a finite sum involving moments of a beta random variable with certain parameters. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 3","pages":"342 - 355"},"PeriodicalIF":1.4,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4556900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}

{"title":"Continuity Criterion for Locally Bounded Automorphisms of Connected Reductive Lie Groups","authors":"A. I. Shtern","doi":"10.1134/S1061920821030080","DOIUrl":"10.1134/S1061920821030080","url":null,"abstract":"<p> We prove that every locally bounded automorphism of a reductive Lie group is continuous. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 3","pages":"356 - 357"},"PeriodicalIF":1.4,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4556901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}

{"title":"Lie Symmetry and Exact Solution of the Time-Fractional Hirota–Satsuma Korteweg–de Vries System","authors":"H.M. Srivastava,&nbsp;H. Mandal,&nbsp;B. Bira","doi":"10.1134/S106192082103002X","DOIUrl":"10.1134/S106192082103002X","url":null,"abstract":"<p> In the present work, we consider the nonlinear time-fractional Hirota-Satsuma KdV (Korteweg-de Vries) system in the sense of the Riemann-Liouville fractional calculus and the Erdélyi-Kober fractional calculus. By appealing to Lie group analysis, we derive the symmetry groups of transformations under which the given equations remain invariant. We also construct the symmetry reductions and particular group invariant solutions for the given system of equations. Finally, in order to highlight the importance of the study, the physical significance of the solution, which is described in this paper, is investigated and illustrated graphically. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"28 3","pages":"284 - 292"},"PeriodicalIF":1.4,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4556909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}