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Том 193, № 2 (2017)
- Год: 2017
- Статей: 11
- URL: https://journals.rcsi.science/0040-5779/issue/view/10447
Article
Local solvability and solution blow-up of one-dimensional equations of the Yajima–Oikawa–Satsuma type
Аннотация
We consider one-dimensional equations of the type of the Yajima–Oikawa–Satsuma ion acoustic wave equation and prove the local solvability. Using the test function method, we obtain sufficient conditions for solution blow-up and estimate the blow-up time.
1561-1573
Phase portraits of the full symmetric Toda systems on rank-2 groups
Аннотация
We continue investigations begun in our previous works where we proved that the phase diagram of the Toda system on special linear groups can be identified with the Bruhat order on the symmetric group if all eigenvalues of the Lax matrix are distinct or with the Bruhat order on permutations of a multiset if there are multiple eigenvalues. We show that the phase portrait of the Toda system and the Hasse diagram of the Bruhat order coincide in the case of an arbitrary simple Lie group of rank 2. For this, we verify this property for the two remaining rank-2 groups, Sp(4,ℝ) and the real form of G2.
1574-1592
Stability of solitary waves in membrane tubes: A weakly nonlinear analysis
Аннотация
We study the problem of the stability of solitary waves propagating in fluid-filled membrane tubes. We consider only waves whose speeds are close to speeds satisfying a linear dispersion relation (it is well known that there can be four families of solitary waves with such speeds), i.e., the waves with small (but finite) amplitudes branching from the rest state of the system. In other words, we use a weakly nonlinear description of solitary waves and show that if the solitary wave speed is bounded from zero, then the solitary wave itself is orbitally stable independently of whether the fluid is in the rest state at the initial time.
1593-1601
Two-dimensional nuclear Coulomb scattering of a slow quantum particle
Аннотация
We study two-dimensional scattering of a quantum particle by the superposition of a Coulomb potential and a central short-range potential. We analyze the low-energy asymptotic behavior of all radial wave functions, partial phases, and scattering cross sections of such a particle. We propose two approaches for evaluating the scattering length and the effective radius.
1602-1629
Rectangular superpolynomials for the figure-eight knot 41
Аннотация
We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot 41 in an arbitrary rectangular representation R = [rs] as a sum over all Young subdiagrams λ of R with surprisingly simple coefficients of the Z factors. Intriguingly, these coefficients are constructed from the quantum dimensions of symmetric representations of the groups SL(r) and SL(s) and restrict the summation to diagrams with no more than s rows and r columns. Moreover, the β-deformation to Macdonald dimensions yields polynomials with positive integer coefficients, which are plausible candidates for the role of superpolynomials for rectangular representations. Both the polynomiality and the positivity of the coefficients are nonobvious, nevertheless true. This generalizes the previously known formulas for symmetric representations to arbitrary rectangular representations. The differential expansion allows introducing additional gradings. For the trefoil knot 31, to which our results for the knot 41 are immediately extended, we obtain the so-called fourth grading of hyperpolynomials. The property of factorization in roots of unity is preserved even in the five-graded case.
1630-1646
Vacuum effects for a one-dimensional “hydrogen atom” with Z > Zcr
Аннотация
For a supercritical Coulomb source with a chargeZ > Zcr in 1+1 dimensions, we study the nonperturbative properties of the vacuum density ρVP(x) and the energy ɛVP. We show that for corresponding problem parameters, nonlinear effects in the supercritical region can lead to behavior of the vacuum energy differing significantly from the perturbative quadratic growth, to the extent of an (almost) quadratic decrease of the form −|η|Z2 into the negative region. We also show that although approaches for calculating vacuum expectations values and the behavior of ρVP(x) in the supercritical region for various numbers of spatial dimensions indeed have many common features, ɛVP for 1+1 dimensions in the supercritical region nevertheless has several specific features determined by the one-dimensionality of the problem.
1647-1674
Multipoint scatterers with bound states at zero energy
Аннотация
We study multipoint scatterers with bound states at zero energy in three-dimensional space. We construct examples of such scatterers with multiple zero eigenvalues or with strong multipole localization of zeroenergy bound states.
1675-1679
A concise review of pseudobosons, pseudofermions, and their relatives
Аннотация
We review some basic definitions and a few facts recently established for D-pseudobosons and pseudofermions. We also discuss an extended version of pseudofermions based on biorthogonal bases in a finitedimensional Hilbert space and describe some examples in detail.
1680-1693
Construction of a set of p-adic distributions
Аннотация
Adapting some methods for real-valued Gibbs measures on Cayley trees to the p-adic case, we construct several p-adic distributions on the set ℤp of p-adic integers. In addition, we give conditions under which these p-adic distributions become p-adic measures (i.e., bounded distributions).
1694-1702
Cylindrically symmetric gravitational-wavelike space–times
Аннотация
We present Noether symmetries of a geodetic Lagrangian for a time-conformal cylindrically symmetric space–time. We introduce a time-conformal factor in the general cylindrically symmetric space–time to make it nonstatic and then find approximate Noether symmetries of the action of the corresponding Lagrangian. Taking the perturbation up to the first order, we find all Lagrangians for cylindrically symmetric space–times for which approximate Noether symmetries exist.
1703-1714
New information-entropic relations for Clebsch–Gordan coefficients
Аннотация
Using properties of the Shannon and Tsallis entropies, we obtain new inequalities for the Clebsch–Gordan coefficients of the group SU(2). For this, we use squares of the Clebsch–Gordan coefficients as probability distributions. The obtained relations are new characteristics of correlations in a quantum system of two spins. We also find new inequalities for Hahn polynomials and the hypergeometric functions 3F2.
1715-1724