Russian Journal of Mathematical Physics最新文献

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.

DOI 10.1134/S1061920823040155

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 (mathbb{Z}^d), (dge1). We study the Cauchy problem and assume that the initial date is a random function. We introduce the family of initial probability measures ({mu_0^varepsilon,varepsilon >0}) slowly varying on the linear scale (1/varepsilon). For times of order (varepsilon^{-kappa}), (0<kappale1), we study the distribution of a random solution and prove the convergence of its covariance to a limit as (varepsilonto0). If (kappa<1), then the limit covariance is time stationary. In the case when (kappa=1), 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.

DOI 10.1134/S1061920823040076

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.

DOI 10.1134/S1061920823040131

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.

DOI 10.1134/S1061920823040052

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.

DOI 10.1134/S1061920823040106

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.

DOI 10.1134/S1061920823040179

In (L_2(mathbb{R}^d)), we consider an elliptic differential operator (mathcal{A}_varepsilon ! = ! - operatorname{div} g(mathbf{x}/varepsilon) nabla + varepsilon^{-2} V(mathbf{x}/varepsilon)), ( varepsilon > 0), with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian (mathcal{A}_varepsilon), analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator (mathcal{A}_1) are studied (the so called high-energy homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in (L_2(mathbb{R}^d))-norm for small (varepsilon) are obtained.

DOI 10.1134/S1061920823040064

The Cauchy problem for a one-dimensional (nonlinear) shallow water equations over a variable bottom (D(x)) is considered in an extended basin bounded from two sides by shores (where the bottom degenerates, (D(a)=0)), 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.

DOI 10.1134/S1061920823040143

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.

DOI 10.1134/S1061920823040015

It is proved that if (G) is a connected solvable group and (pi) is a (not necessarily continuous) representation of (G) in a finite-dimensional vector space (E), then there is a basis in (E) in which the matrices of the representation operators of (pi) have upper triangular form. The assertion is extended to connected solvable locally compact groups (G) having a connected normal subgroup for which the quotient group is a Lie group.

DOI 10.1134/S1061920823040180