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Trace Formulas for a Complex KdV Equation 复杂 KdV 方程的示踪公式
IF 1.7 3区 物理与天体物理
Russian Journal of Mathematical Physics Pub Date : 2024-03-19 DOI: 10.1134/S106192084010096
E. Korotyaev
{"title":"Trace Formulas for a Complex KdV Equation","authors":"E. Korotyaev","doi":"10.1134/S106192084010096","DOIUrl":"10.1134/S106192084010096","url":null,"abstract":"<p> Faddeev and Zakharov determined the trace formulas for the KdV equation with real initial conditions in 1971. We reprove these results for the KdV equation with complex initial conditions. The Lax operator is a Schrödinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine series of trace formulas. Here we have a new term: a singular measure, which is absent for real potentials. Moreover, we estimate of sum of the imaginary part of eigenvalues plus the singular measure in terms of the norm of potentials. The proof is based on classical results about the Hardy spaces. </p><p> <b> DOI</b> 10.1134/S106192084010096 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"31 1","pages":"112 - 131"},"PeriodicalIF":1.7,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140171706","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
On Solutions of the Navier Problem for a Polyharmonic Equation in Unbounded Domains 论无界域多谐方程的纳维问题解法
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/S1061920823040209
H.A. Matevossian
{"title":"On Solutions of the Navier Problem for a Polyharmonic Equation in Unbounded Domains","authors":"H.A. Matevossian","doi":"10.1134/S1061920823040209","DOIUrl":"10.1134/S1061920823040209","url":null,"abstract":"<p> The polyharmonic Navier problem is considered, the uniqueness (non-uniqueness) of its solution is studied in unbounded domains under the assumption that the generalized solution of this problem has a finite Dirichlet integral with weight <span>(|x|^a)</span>. Depending on the values of the parameter <span>(a)</span>, uniqueness theorems are proved and exact formulas are found for calculating the dimension of the space of solutions of the Navier problem for a polyharmonic equation in the exterior of a compact set and in a half-space. </p><p> <b> DOI</b> 10.1134/S1061920823040209 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"713 - 716"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056503","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
Averaging Method for Quasi-Linear Hyperbolic Systems 准线性双曲系统的平均法
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/S1061920823040118
V.B. Levenshtam
{"title":"Averaging Method for Quasi-Linear Hyperbolic Systems","authors":"V.B. Levenshtam","doi":"10.1134/S1061920823040118","DOIUrl":"10.1134/S1061920823040118","url":null,"abstract":"<p> The paper considers the Cauchy problem for a multidimensional quasilinear hyperbolic system of differential equations with the data rapidly oscillating in time. This data do not explicitly depend on spatial variables. The method by N. M. Krylov–N. N. Bogolyubov is developed and justified for these systems. Also an algorithm is developed and justified, based on this method and the method of two-scale expansions, for constructing the complete asymptotics of solutions. </p><p> <b> DOI</b> 10.1134/S1061920823040118 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"552 - 560"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056592","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
Trace of the Resolvent of the Laplace Operator on a Metric Graph 公制图上拉普拉斯算子残差的轨迹
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/S1061920823040192
A. A. Tolchennikov
{"title":"Trace of the Resolvent of the Laplace Operator on a Metric Graph","authors":"A. A. Tolchennikov","doi":"10.1134/S1061920823040192","DOIUrl":"10.1134/S1061920823040192","url":null,"abstract":"<p> In the paper, using Krein’s resolvent formula, we find an asymptotics of the resolvent of the trace of the Laplace operator on a metric graph. </p><p> <b> DOI</b> 10.1134/S1061920823040192 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"704 - 712"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056386","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
Method of Potential Operators for Interaction Problems on Unbounded Hypersurfaces in (mathbb{R}^{n}) for Dirac Operators 针对狄拉克算子的 $$mathbb{R}^{n}$ 中无边界超曲面上相互作用问题的势算子方法
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/S1061920823040167
V. S. Rabinovich
{"title":"Method of Potential Operators for Interaction Problems on Unbounded Hypersurfaces in (mathbb{R}^{n}) for Dirac Operators","authors":"V. S. Rabinovich","doi":"10.1134/S1061920823040167","DOIUrl":"10.1134/S1061920823040167","url":null,"abstract":"<p> We consider the <span>(L_{p})</span>-theory of interaction problems associated with Dirac operators with singular potentials of the form <span>(D=mathfrak{D}_{m,Phi }+Gammadelta_{Sigma})</span> where </p><p> is a Dirac operator on <span>(mathbb{R}^{n})</span>, <span>(alpha_{1},alpha_{2},dots,alpha _{n},alpha_{n+1})</span> are Dirac matrices, <span>(m)</span> is a variable mass, <span>(Phi mathbb{I}_{N})</span> electrostatic potential, <span>(Gammadelta_{Sigma})</span> is a singular potential with support on smooth hypersurfaces <span>(Sigma subsetmathbb{R}^{n}.)</span> </p><p> We associate with the formal Dirac operator <span>(D)</span> the interaction (transmission) problem on <span>(mathbb{R}^{n}diagdownSigma)</span> with the interaction conditions on <span>(Sigma)</span>. Applying the method of potential operators we reduce the interaction problem to a pseudodifferential equation on <span>(Sigma.)</span> The main aim of the paper is the study of Fredholm property of these pseudodifferential operators on unbounded hypersurfaces <span>(Sigma)</span> and applications to the study of Fredholmness of interaction problems on unbounded smooth hypersurfaces in Sobolev and Besov spaces. </p><p> <b> DOI</b> 10.1134/S1061920823040167 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"674 - 690"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056431","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
On Degenerate Orbits of Real Lie Algebras in Multidimensional Complex Spaces 论多维复杂空间中实列代数的退化轨道
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/S1061920823040027
A.V. Atanov, A.V. Loboda
{"title":"On Degenerate Orbits of Real Lie Algebras in Multidimensional Complex Spaces","authors":"A.V. Atanov,&nbsp;A.V. Loboda","doi":"10.1134/S1061920823040027","DOIUrl":"10.1134/S1061920823040027","url":null,"abstract":"<p>The article studies (locally) holomorphically homogeneous real hypersurfaces of complex spaces. Currently, the problem of classifying such hypersurfaces is completely solved only in the spaces <span>(mathbb{C}^{2})</span> and <span>(mathbb{C}^{3})</span>. As the dimension of the ambient space grows, so does the relative part of Levi-degenerate manifolds in the family of all homogeneous hypersurfaces. In particular, this family includes holomorphically degenerate hypersurfaces, which are (locally) direct products of homogeneous hypersurfaces from spaces of smaller dimensions and spaces <span>(mathbb{C}^{k})</span>. The article proves a sufficient Levi-degeneracy condition of all orbits in spaces <span>(mathbb{C}^{n+1})</span> <span>((n ge 3))</span> for <span>((2n+1))</span>-dimensional Lie algebras of holomorphic vector fields having full rank at the points in <span>(mathbb{C}^{n+1})</span>. The proven condition is the existence of an abelian subalgebra of codimension 2 in the Lie algebra under discussion. It is shown that in the case <span>(n = 3)</span>, this condition holds for a large family of 7-dimensional Lie algebras. Holomorphically homogeneous hypersurfaces, i.e. the orbits of these algebras in <span>(mathbb{C}^{4})</span> can only be Levi-degenerate manifolds. We provide an example of a family of 7-dimensional Lie algebras that have 5-dimensional abelian ideals and Levi-degenerate (but not holomorphically degenerate) orbits. </p><p> <b> DOI</b> 10.1134/S1061920823040027 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"432 - 442"},"PeriodicalIF":1.7,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056496","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
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
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