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Classification of Singularities of the Liouville Foliation of an Integrable Elliptical Billiard with a Potential of Fourth Degree 具有四阶势能的可积分椭圆台球的柳维尔奇点分类
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040155
S.E. Pustovoitov
{"title":"Classification of Singularities of the Liouville Foliation of an Integrable Elliptical Billiard with a Potential of Fourth Degree","authors":"S.E. Pustovoitov","doi":"10.1134/S1061920823040155","DOIUrl":"10.1134/S1061920823040155","url":null,"abstract":"<p> The paper is devoted to the study of a billiard bounded by an ellipse and equipped with a fourth degree potential as an integrable Hamiltonian system with two degrees of freedom. In previous works, the author described the structure of the Liouville foliation of such a system on nonsingular levels of the Hamiltonian in terms of Fomenko–Zieschang invariants: marked molecules and 3-atoms. Moreover, the dependence of the structure of the bifurcation diagram on the parameters of the potential has been established. The present work continues this study. Thus, the structure of the Liouville foliation in a neighborhood of critical layers containing a nondegenerate singular point of rank 0 or a degenerate orbit has been described. A classification of the obtained semilocal singularities was given. Finally, connections of our system with well-known cases of rigid body dynamics containing equivalent singularities is established. </p><p> <b> DOI</b> 10.1134/S1061920823040155 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"643 - 673"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056626","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}
引用次数: 0
Transport Equation for the Harmonic Crystal Coupled to a Klein–Gordon Field 与克莱因-戈登场耦合的谐波晶体的传输方程
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040076
T.V. Dudnikova
{"title":"Transport Equation for the Harmonic Crystal Coupled to a Klein–Gordon Field","authors":"T.V. Dudnikova","doi":"10.1134/S1061920823040076","DOIUrl":"10.1134/S1061920823040076","url":null,"abstract":"<p> We consider the Hamiltonian system consisting of the Klein–Gordon field coupled to an infinite harmonic crystal. The dynamics of the coupled system is translation-invariant with respect to the space translations in <span>(mathbb{Z}^d)</span>, <span>(dge1)</span>. We study the Cauchy problem and assume that the initial date is a random function. We introduce the family of initial probability measures <span>({mu_0^varepsilon,varepsilon &gt;0})</span> slowly varying on the linear scale <span>(1/varepsilon)</span>. For times of order <span>(varepsilon^{-kappa})</span>, <span>(0&lt;kappale1)</span>, we study the distribution of a random solution and prove the convergence of its covariance to a limit as <span>(varepsilonto0)</span>. If <span>(kappa&lt;1)</span>, then the limit covariance is time stationary. In the case when <span>(kappa=1)</span>, the covariance changes in time and is governed by a semiclassical transport equation. We give an application to the case of the Gibbs initial measures. </p><p> <b> DOI</b> 10.1134/S1061920823040076 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"501 - 521"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056668","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}
引用次数: 0
Asymptotics of the Whispering Gallery-Type in the Eigenproblem for the Laplacian in a Domain of Revolution Diffeomorphic To a Solid Torus 实心圆环差分革命域中拉普拉斯函数特征问题中的 "悄悄话画廊 "渐近论
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040131
D.S. Minenkov, S.A. Sergeev
{"title":"Asymptotics of the Whispering Gallery-Type in the Eigenproblem for the Laplacian in a Domain of Revolution Diffeomorphic To a Solid Torus","authors":"D.S. Minenkov,&nbsp;S.A. Sergeev","doi":"10.1134/S1061920823040131","DOIUrl":"10.1134/S1061920823040131","url":null,"abstract":"<p> We consider the eigenproblem for the Laplacian inside a three-dimensional domain of revolution diffeomorphic to a solid torus, and construct asymptotic eigenvalues and eigenfunctions (quasimodes) of the whispering gallery-type. The whispering gallery-type asymptotics are localized near the boundary of the domain, and an explicit analytic representation in terms of Airy functions is constructed for such asymptotics. There are several different scales in the problem, which makes it possible to apply the procedure of adiabatic approximation in the form of operator separation of variables to reduce the initial problem to one-dimensional problems up to a small correction. We also discuss the relationship between the constructed whispering gallery-type asymptotics and classical billiards in the corresponding domain, in particular, such asymptotics correspond to almost integrable billiards with proper degeneracy. We illustrate the results in the case when a domain of revolution is obtained by the rotation of a triangle with rounded wedges. </p><p> <b> DOI</b> 10.1134/S1061920823040131 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"599 - 620"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056495","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}
引用次数: 0
Asymptotics of the Cauchy Problem for the One-Dimensional Schrödinger Equation with Rapidly Oscillating Initial Data and Small Addition to the Smooth Potential 一维薛定谔方程的考希问题的渐近性与快速振荡初始数据和光滑势的微小附加值
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040052
S. Yu. Dobrokhotov
{"title":"Asymptotics of the Cauchy Problem for the One-Dimensional Schrödinger Equation with Rapidly Oscillating Initial Data and Small Addition to the Smooth Potential","authors":"S. Yu. Dobrokhotov","doi":"10.1134/S1061920823040052","DOIUrl":"10.1134/S1061920823040052","url":null,"abstract":"<p> We study the asymptotic solution of the Cauchy problem with rapidly changing initial data for the one-dimensional nonstationary Schrödinger equation with a smooth potential perturbed by a small rapidly oscillating addition. Solutions to such a Cauchy problem are described by moving, rapidly oscillating wave packets. According to long-standing results of V.S. Buslaev and S.Yu. Dobrokhotov, the construction of a solution to this problem can be constructed applying the sequential use of the adiabatic and semiclassical approximations. In the general situation, the construction the asymptotic formula reduces to solving a large number of auxiliary spectral problems for families of Bloch functions of ordinary differential operators of Sturm–Liouville type, and the answer is presented in an ineffective form. On the other hand, the assumption that the rapidly oscillating perturbation of the potential is small gives the opportunity, firstly, to write asymptotic formulas for solutions of the indicated auxiliary spectral problems and, secondly, to save, in the construction of the answer to the original problem, only finitely many these problems and their solutions. Bounds are obtained for problem parameters answering when such considerations can be implemented and, if the corresponding conditions on the parameters are satisfied, asymptotic solutions are constructed. </p><p> <b> DOI</b> 10.1134/S1061920823040052 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"466 - 479"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056502","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}
引用次数: 0
Nonlinear Long Standing Waves with Support Bounded by Caustics or Localized in the Vicinity of a Two-Link Trajectory 非线性长驻波，其支撑点受凹陷约束或在双链轨迹附近局部化
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040106
A.I. Klevin, A.V. Tsvetkova
{"title":"Nonlinear Long Standing Waves with Support Bounded by Caustics or Localized in the Vicinity of a Two-Link Trajectory","authors":"A.I. Klevin,&nbsp;A.V. Tsvetkova","doi":"10.1134/S1061920823040106","DOIUrl":"10.1134/S1061920823040106","url":null,"abstract":"<p> The paper is devoted to describing the dynamics and uprush of time-periodic long waves in basins with gentle shores. We consider waves that are defined by solutions localized between caustics in the domain bounded by the shores of the basin. We also consider solutions localized in the vicinity of a periodic trajectory which, during the period, has exactly two intersections with the boundary of such a domain. </p><p> <b> DOI</b> 10.1134/S1061920823040106 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"543 - 551"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056557","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}
引用次数: 0
Estimation of the Approximation of Continuous Periodic Functions by Fourier Sums 用傅里叶和估计连续周期函数的近似值
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040179
T.Yu. Semenova
{"title":"Estimation of the Approximation of Continuous Periodic Functions by Fourier Sums","authors":"T.Yu. Semenova","doi":"10.1134/S1061920823040179","DOIUrl":"10.1134/S1061920823040179","url":null,"abstract":"<p> An asymptotically exact estimate for the norm of the difference between a function and the partial sum of its Fourier series is obtained in terms of the modulus of continuity of the function. The values of the modulus of continuity of the argument that are less than the optimal one are considered. </p><p> <b> DOI</b> 10.1134/S1061920823040179 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"691 - 700"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056858","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}
引用次数: 0
High-Energy Homogenization of a Multidimensional Nonstationary Schrödinger Equation 多维非稳态薛定谔方程的高能量均质化
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040064
M. Dorodnyi
{"title":"High-Energy Homogenization of a Multidimensional Nonstationary Schrödinger Equation","authors":"M. Dorodnyi","doi":"10.1134/S1061920823040064","DOIUrl":"10.1134/S1061920823040064","url":null,"abstract":"<p> In <span>(L_2(mathbb{R}^d))</span>, we consider an elliptic differential operator <span>(mathcal{A}_varepsilon ! = ! - operatorname{div} g(mathbf{x}/varepsilon) nabla + varepsilon^{-2} V(mathbf{x}/varepsilon))</span>, <span>( varepsilon &gt; 0)</span>, with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian <span>(mathcal{A}_varepsilon)</span>, analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator <span>(mathcal{A}_1)</span> are studied (the so called high-energy homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in <span>(L_2(mathbb{R}^d))</span>-norm for small <span>(varepsilon)</span> are obtained. </p><p> <b> DOI</b> 10.1134/S1061920823040064 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"480 - 500"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056384","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}
引用次数: 0
Asymptotics of Long Nonlinear Propagating Waves in a One-Dimensional Basin with Gentle Shores 具有平缓海岸的一维盆地中长非线性传播波的渐近学
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040143
D.S. Minenkov, M.M. Votiakova
{"title":"Asymptotics of Long Nonlinear Propagating Waves in a One-Dimensional Basin with Gentle Shores","authors":"D.S. Minenkov,&nbsp;M.M. Votiakova","doi":"10.1134/S1061920823040143","DOIUrl":"10.1134/S1061920823040143","url":null,"abstract":"<p> The Cauchy problem for a one-dimensional (nonlinear) shallow water equations over a variable bottom <span>(D(x))</span> is considered in an extended basin bounded from two sides by shores (where the bottom degenerates, <span>(D(a)=0)</span>), or by a shore and a wall. The short-wave asymptotics of the linearized system in the form of a propagating localized wave is constructed. After applying to the constructed functions a simple parametric or explicit change of variables proposed in recent papers (Dobrokhotov, Minenkov, Nazaikinsky, 2022 and Dobrokhotov, Kalinichenko, Minenkov, Nazaikinsky, 2023), we obtain the asymptotics of the original nonlinear problem. On the constructed families of functions, the ratio of the amplitude and the wavelength is studied for which hte wave does not collapse when running up to the shore. </p><p> <b> DOI</b> 10.1134/S1061920823040143 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"621 - 642"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056586","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}
引用次数: 0
Elementary Differential Singularities of Three-Dimensional Nijenhuis Operators 三维尼延胡斯算子的初等微分奇异性
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040015
D. Akpan, A. Oshemkov
{"title":"Elementary Differential Singularities of Three-Dimensional Nijenhuis Operators","authors":"D. Akpan,&nbsp;A. Oshemkov","doi":"10.1134/S1061920823040015","DOIUrl":"10.1134/S1061920823040015","url":null,"abstract":"<p> In the paper, three-dimensional Nijenhuis operators are studied that have differential singularities, i.e., such points at which the coefficients of the characteristic polynomials are dependent. The case is studied in which the differentials of all invariants of the Nijenhuis operator are proportional, as well as the case when two invariants are functionally independent and the third defines a fold-type singularity. In particular, new examples of three-dimensional Nijenhuis operators with singularities of the specified type are constructed. </p><p> <b> DOI</b> 10.1134/S1061920823040015 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"425 - 431"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056778","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}
引用次数: 0
Lie’s Theorem for Solvable Connected Lie Groups Without the Continuity Assumption 无连续性假设的可解连通李群的李氏定理
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI: 10.1134/S1061920823040180
A. I. Shtern
{"title":"Lie’s Theorem for Solvable Connected Lie Groups Without the Continuity Assumption","authors":"A. I. Shtern","doi":"10.1134/S1061920823040180","DOIUrl":"10.1134/S1061920823040180","url":null,"abstract":"<p> It is proved that if <span>(G)</span> is a connected solvable group and <span>(pi)</span> is a (not necessarily continuous) representation of <span>(G)</span> in a finite-dimensional vector space <span>(E)</span>, then there is a basis in <span>(E)</span> in which the matrices of the representation operators of <span>(pi)</span> have upper triangular form. The assertion is extended to connected solvable locally compact groups <span>(G)</span> having a connected normal subgroup for which the quotient group is a Lie group. </p><p> <b> DOI</b> 10.1134/S1061920823040180 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"701 - 703"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056387","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}
引用次数: 0
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