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 Russian Journal of Mathematical PhysicsJournal Prestige (SJR): 0.696 Citation Impact (citeScore): 1Number of Followers: 0      Subscription journal ISSN (Print) 1061-9208 - ISSN (Online) 1555-6638 Published by Springer-Verlag  [2469 journals]
• Continuity Criterion for Locally Bounded Automorphisms of Central
Extensions of Perfect Lie Groups with Discrete Center

Abstract: We prove that every locally bounded automorphism of a connected not necessarily linear Lie central extension of a connected perfect Lie group with discrete center is continuous if and only if it is continuous on the center.
PubDate: 2022-03-01

• On a Nonlinear Nonlocal Parabolic Problem

Abstract: We consider the Cauchy problem for a parabolic equation with a Ё-Laplacian or a general second-order quasilinear equation with boundary conditions of the Bitsadze–Samarskii type. We prove that at least one generalized solution of such problem exists.
PubDate: 2022-03-01

• Integrable Hamiltonian Systems on the Symplectic Realizations of
$$\textbf{e}(3)^*$$

Abstract: The phase space of a gyrostat with a fixed point and a heavy top is the Lie–Poisson space $$\textbf{e}(3)^*\cong \mathbb{R}^3\times \mathbb{R}^3$$ dual to the Lie algebra $$\textbf{e}(3)$$ of the Euclidean group $$E(3)$$ . One has three naturally distinguished Poisson submanifolds of $$\textbf{e}(3)^*$$ : (i) the dense open submanifold $$\mathbb{R}^3\times \dot{\mathbb{R}}^3\subset \textbf{e}(3)^*$$ which consists of all $$4$$ -dimensional symplectic leaves ( $$\vec{\Gamma}^2>0$$ ); (ii) the $$5$$ -dimensional Poisson submanifold of $$\mathbb{R}^3\times \dot{\mathbb{R}}^3$$ defined by $$\vec{J}\cdot \vec{\Gamma} = \mu \vec{\Gamma}$$ ; (iii) the $$5$$ -dimensional Poisson submanifold of $$\mathbb{R}^3\times \dot{\mathbb{R}}^3$$ defined by $$\vec{\Gamma}^2 = \nu^2$$ , where $$\dot{\mathbb{R}}^3:= \mathbb{R}^3\backslash \{0\}$$ , $$(\vec{J}, \vec{\Gamma})\in \mathbb{R}^3\times \mathbb{R}^3\cong \textbf{e}(3)^*$$ and $$\nu < 0$$ , $$\mu$$ are some fixed real parameters. Using the $$U(2,2)$$ -invariant symplectic structure of Penrose twistor space we find full and complete $$E(3)$$ -equivariant symplectic realizations of these Poisson submanifolds which are $$8$$ -dimensional for (i) and $$6$$ -dimensional for (ii) and (iii). As a consequence of the above, Hamiltonian systems on $$\textbf{e}(3)^*$$ lift to Hamiltonian systems on the above symplectic realizations. In this way, after lifting the integrable cases of a gyrostat with a fixed point and of a heavy top, we obtain a large family of integrable Hamiltonian systems on the phase spaces defined by these symplectic realizations.
PubDate: 2022-03-01

• Uniform Asymptotics in the Form of Airy Functions for Bound States of the
Quantum Anisotropic Kepler Problem Localized in a Neighborhood of Annuli

Abstract: The joint spectrum of the Schrödinger operator of an anisotropic Kepler problem (the component along one of the axes of the diagonal mass tensor differs from the components along the other two axes) and the angular momentum operator are considered. Using the theory of Maslov’s complex germ, series of corresponding semiclassical stationary states (with complex phases) localized in the vicinity of flat disks are constructed. The wave functions of these states, in the direction normal to the plane of the disk, have the form of Hermite–Gaussian functions; in the direction of the polar angle coordinate in the plane of the disk, they oscillate harmonically; in the radial direction in the plane of the disk, their behavior is described by the Airy function of a composed argument: inside the disk, the wave functions oscillate and, outside it, decay. The existence of such states is due to the existence of Floquet solutions of certain differential equations with periodic coefficients.
PubDate: 2022-03-01

• Uniformly and Locally Convex Asymmetric Spaces

Abstract: For uniformly convex asymmetric spaces, questions concerning nonempty intersections of a nested system of bounded convex closed sets are considered. Questions concerning the density of sets of points of existence and approximative uniqueness are studied in these spaces for the case of nonempty closed subsets.
PubDate: 2022-03-01

• Maslov’s Complex Germ in the Cauchy Problem for a Wave Equation with
a Jumping Velocity

Abstract: Using Maslov’s complex germ in the Cauchy problem for a wave equation, we consider the asymptotics of the solution of the Cauchy problem in which the velocity depends irregularly on a small parameter.
PubDate: 2022-03-01

• On Exceptional Quadrics

Abstract: It is proved that the graded Lie algebras of infinitesimal holomorphic automorphisms of a nondegenerate quadric of codimension $$k$$ do not have weight components more than $$2k$$ . It is also proved that, for $$k \leq 3$$ , there are no graded components of weight greater than 2. Several questions are formulated.
PubDate: 2022-03-01

• Distribution of the Spectrum of the Airy Operator Perturbed by Delta
Interactions

Abstract: In the paper, the asymptotics of eigenvalues of a perturbed Airy operator is obtained.
PubDate: 2022-03-01

• On the Inverse Semigroup of Bimodules over a $$C^*$$ -Algebra

Abstract: It was noticed recently that, given a metric space $$(X,d_X)$$ , the equivalence classes of metrics on the disjoint union of the two copies of $$X$$ coinciding with $$d_X$$ on each copy form an inverse semigroup $$M(X)$$ with respect to concatenation of metrics. Now put this inverse semigroup construction in a more general context, namely, we define, for a $$C^*$$ -algebra $$A$$ , an inverse semigroup $$S(A)$$ of Hilbert $$C^*$$ - $$A$$ - $$A$$ -bimodules. When $$A$$ is the uniform Roe algebra $$C^*_u(X)$$ of a metric space $$X$$ , we construct a mapping $$M(X)\to S(C^*_u(X))$$ and show that this mapping is injective, but not surjective in general. This allows to define an analog of the inverse semigroup $$M(X)$$ that does not depend on the choice of a metric on $$X$$ within its coarse equivalence class.
PubDate: 2022-03-01

• Asymptotic Solutions of the Cauchy Problem for the Nonlinear Shallow Water
Equations in a Basin with a Gently Sloping Beach

Abstract: Small amplitude solutions of the nonlinear shallow water equations in a one- or two-dimensional domain are considered. The amplitude is characterized by a small parameter $$\varepsilon$$ . It is assumed that the basin depth is a smooth function whose gradient is nowhere zero on the set of its zeros (i.e., on the coastline in the absence of waves). A solution of the equations is understood to be a triple (time-dependent domain, free surface elevation, horizontal velocity) smoothly depending on $$\varepsilon$$ and such that (i) the free surface elevation and the horizontal velocity are zero for $$\varepsilon =0$$ ; (ii) the sum of the free surface elevation and the depth is positive in the domain and zero on the boundary; (iii) the free surface elevation and the horizontal velocity are smooth in the closed domain and satisfy the equations there. An asymptotic solution modulo $$O( \varepsilon ^N)$$ is defined in a similar way except that the equations must be satisfied modulo $$O( \varepsilon ^N)$$ . We prove that, in this setting, the nonlinear shallow water equations with small smooth initial data have an asymptotically unique asymptotic solution modulo $$O( \varepsilon ^N)$$ for arbitrary $$N$$ . The proof is constructive (and leads to simple explicit formulas for the leading asymptotic term). The construction uses a change of variables (depending on the unknown solution and resembling the Carrier–Greenspan transformation) that maps the unknown varying domain onto the unperturbed domain. The resulting nonlinear system is within the scope of regular perturbation theory. The zero approximation is a Cauchy problem for a linear hyperbolic system with degeneracy on the boundary, whose unique solvability in the class of smooth functions is proved by lifting the problem to a closed 3-manifold (where the spatial part of the operator turns out to be hypoelliptic).
PubDate: 2022-03-01

• Contrast Structures in the Reaction-Diffusion-Advection Problem in the
Case of a Weak Reaction Discontinuity

Abstract: In the paper, a boundary value problem for a singularly perturbed equation of reaction-diffusion-advection in a two-dimensional domain is considered. The key feature of the problem under consideration is the weak discontinuity of the reactive term. The discontinuity occurs on a simple closed curve known in advance that lies entirely inside the domain. It is shown that such a problem has a solution with an inner transition layer localized near the discontinuity curve. For this solution, an asymptotic expansion with respect to a small parameter is constructed, and existence theorems are proved, together with Lyapunov asymptotic stability. As a method of the proof, the asymptotic method of differential inequalities is used. An example is given illustrating the constructions carried out in the paper.
PubDate: 2022-03-01

• On Some Degenerate Differential and Degenerate Difference Operators

Abstract: The aim of this paper is to make use of certain degenerate differential and degenerate difference operators in order to study some identities involving the degenerate harmonic numbers, certain finite sums of general nature, the sums of the values of the generalized falling factorials at consecutive positive integers, and the degenerate Laguerre polynomials.
PubDate: 2022-03-01

• Structurally Stable Nondegenerate Singularities of Integrable Systems

Abstract: In this paper, we study singularities of the Lagrangian fibration given by a completely integrable system. We prove that a nondegenerate singular fiber satisfying the so-called connectedness condition is structurally stable under (small enough) real-analytic integrable perturbations of the system. In other words, the topology of the fibration in a neighborhood of such a fiber is preserved after any such perturbation. As an illustration, we show that a simple saddle-saddle singularity of the Kovalevskaya top is structurally stable under real-analytic integrable perturbations and not structurally stable under $$C^\infty$$ -smooth integrable perturbations.
PubDate: 2022-03-01

• Continuity Criteria for Locally Bounded Automorphisms of Central
Extensions of Perfect Lie Groups

Abstract: 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 $$Z$$ is a connected Abelian group without nontrivial compact subgroups, $$H$$ is a connected perfect Lie group and the short sequence of Lie groups $$\{e\}\to Z\to G\to H\to\{e\}$$ is exact, then every locally bounded automorphism of $$G$$ is continuous if and only if it is continuous on the center of $$G$$ .
PubDate: 2021-10-01
DOI: 10.1134/S1061920821040117

• Analytic Solutions of the Cylindrical Heat Equation with a Heat Source

Abstract: In this article, the superposition and the separation of variables methods are applied in order to investigate the analytical solutions of a heat conduction equation in cylindrical coordinates. The structures of the transient temperature and the heat transfer distributions are summed up for a direct mix of the results of the Fourier–Bessel series of the exponential type for the partial differential equation which we investigate here. Relevant connections of the results, which we have presented in this article, with those in some other closely-related earlier works are also indicated.
PubDate: 2021-10-01
DOI: 10.1134/S1061920821040129

• Complete Integrability of a New Class of Hamiltonian Hydrodynamic Type
Systems

Abstract: 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.
PubDate: 2021-10-01
DOI: 10.1134/S1061920821040099

• Pólya–Carlson Dichotomy for Dynamical Zeta Functions and a Twisted
Burnside–Frobenius Theorem

Abstract: 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.
PubDate: 2021-10-01
DOI: 10.1134/S1061920821040051

• On the Correspondence between the Variational Principles in the Eulerian
and Lagrangian Descriptions

Abstract: 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.
PubDate: 2021-10-01
DOI: 10.1134/S1061920821040014

• The Regularized Free Fall I. Index Computations

Abstract: 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 $$\mu^{\rm CZ}$$ . We consider the nonlocal Lagrangian action functional $${\mathcal{B}}$$ 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 $$x$$ of the Kepler problem. In this article, we look at period 1 only and at dimension one (gravitational free fall). Via a nonlocal Legendre transform regularized periodic Kepler orbits $$x$$ can be interpreted as periodic solutions $$(x,y)$$ of a Hamiltonian delay equation. In particular, regularized $$1$$ -periodic solutions of the free fall are represented variationally in two ways: as critical points $$x$$ of a nonlocal Lagrangian action functional and as critical points $$(x,y)$$ of a nonlocal Hamiltonian action functional. As critical points of the Lagrangian action, the $$1$$ -periodic solutions have a finite Morse index which we compute first. As critical points of the Hamiltonian action $${\mathcal{A}} _ {\mathcal{H}}$$ , one encounters the obstacle, due to nonlocality, that the $$1$$ -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 $${\mathcal{A}} _ {\mathcal{H}}$$ 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.
PubDate: 2021-10-01
DOI: 10.1134/S1061920821040063

• Bifurcations of Essential Spectra Generated by a Small Non-Hermitian Hole.
I. Meromorphic Continuations

Abstract: This paper focuses on bifurcations that occur in the essential spectrum of certain non-Hermitian operators. We consider the eigenvalue problem for a self-adjoint elliptic differential operator in a multidimensional tube-like domain which is infinite along one dimension and can be bounded or unbounded in other dimensions. This self-adjoint eigenvalue problem is perturbed by a small hole cut out of the domain. The boundary of the hole is described by a non-Hermitian Robin-type boundary condition. The main result of the present paper states the existence and describes the properties of local meromorphic continuations of the resolvent of the operator in question through the essential spectrum. The continuations are constructed near the edge of the spectrum and in the vicinity of certain internal threshold points of the spectrum. Then we define the eigenvalues and resonances of the operator as the poles of these continuations and prove that both the edge and the internal thresholds bifurcate into eigenvalues and/or resonances. The total multiplicity of the eigenvalues and resonances bifurcating from internal thresholds can be up to twice larger than the multiplicity of the thresholds. In other words, the perturbation can increase the total multiplicity.
PubDate: 2021-10-01
DOI: 10.1134/S1061920821040026

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