Abstract It is known that the transfer-matrix eigenvalues of the isotropic open Heisenberg quantum spin-1/2 chain with nondiagonal boundary magnetic fields satisfy a \(TQ\) equation with an inhomogeneous term. We derive a discrete Wronskian-type formula relating a solution of this inhomogeneous \(TQ\) equation to the corresponding solution of a dual inhomogeneous \(TQ\) equation.

Abstract To obtain a generating function of the most general form for Hurwitz numbers with an arbitrary base surface and arbitrary ramification profiles, we consider a matrix model constructed according to a graph on an oriented connected surface \(\Sigma\) with no boundary. The vertices of this graph, called stars, are small discs, and the graph itself is a clean dessin d’enfants. We insert source

Abstract We construct three nonequivalent gradings in the algebra \(D_4\simeq so(8)\). The first is the standard grading obtained with the Coxeter automorphism \(C_1=S_{\alpha_2}S_{\alpha_1}S_{\alpha_3}S_{\alpha_4}\) using its dihedral realization. In the second, we use \(C_2=C_1R\), where \(R\) is the mirror automorphism. The third is \(C_3=S_{\alpha_2}S_{\alpha_1}T\), where \(T\) is the external

Abstract As is known, in the reflection equation (RE) algebra associated with an involutive or Hecke \(R\)-matrix, the elements \( \operatorname{Tr} _RL^k\) (called quantum power sums) are central. Here, \(L\) is the generating matrix of this algebra, and \( \operatorname{Tr} _R\) is the operation of taking the \(R\)-trace associated with a given \(R\)-matrix. We consider the problem of whether this

Abstract Using the approach developed by Gautam and Toledano Laredo, we introduce analogues of the category \( \mathfrak{O} \) for representations of the Yangian \(Y_\hbar(A(m,n))\) of a special linear Lie superalgebra and the quantum loop superalgebra \(U_q(LA(m,n))\). We investigate the relation between them and conjecture that these categories are equivalent.

Abstract We realize an example of induced dynamics using new multiplicative determinant relations whose roots give the particle positions. We present both a general scheme for describing completely integrable dynamical systems parameterized by an arbitrary \(N{\times}N\) matrix of momenta and an explicit model that interpolates between the Calogero–Moser and Ruijsenaars–Schneider hyperbolic systems

Abstract We show that both the dKP hierarchy and its strict version can be extended to a wider class of deformations satisfying a larger set of Lax equations. We prove that both extended hierarchies have appropriate linearizations allowing a geometric construction of their solutions.

Abstract We construct a family of determinant representations for scalar products of Bethe vectors in models with \( \mathfrak{gl} (3)\) symmetry. This family is defined by a single generating function containing arbitrary complex parameters but is independent of their specific values. Choosing these parameters in different ways, we can obtain different determinant representations.

Abstract We study elliptic solutions of the semidiscrete B-version of the Kadomtsev–Petviashvili equation and derive the equations of motion of their poles. The auxiliary linear problems for the wave function are the main technical tool.

Abstract We consider nonlocal symmetries that all or all even (all odd) equations of the AKNS hierarchy have. We construct examples of solutions simultaneously satisfying several nonlocal equations of the AKNS hierarchy. We present a detailed study of single-phase solutions.

Abstract The concept of extended Hamiltonian systems allows a geometric interpretation of several integrable and superintegrable systems with polynomial first integrals of a degree depending on a rational parameter. Until now, the extension procedure has been applied only in the case of natural Hamiltonians. We give several examples of application to nonnatural Hamiltonians, such as the Hamiltonian

AbstractWe study properties of the space \(\boldsymbol{\Omega}\) of solutions of a special double confluent Heun equation closely related to the model of a overdamped Josephson junction. We describe operators acting on \(\boldsymbol{\Omega}\) and relations in the algebra \(\mathcal{A}\) generated by them over the real number field. The structure of \(\mathcal{A}\) depends on parameters. We give conditions

AbstractThe recently proposed Kameyama–Nawata–Tao–Zhang (KNTZ ) trick completed the long search for exclusive Racah matrices \( \kern2.3pt\overline{\vphantom{S}\kern3.70831pt}\kern-6.00832pt{S} \) and \(S\) for all rectangular representations. The success of this description is a remarkable achievement of modern knot theory and classical representation theory, which was initially considered a tool

AbstractWe show that the system of nonlinear equations of two-photon propagation of light with real amplitudes of the envelopes can be solved in general form by the classical Liouville method. This system, like other similar systems of Darboux-integrable equations, is related to the modified Liouville equation, and the found solution also provides general solutions of such modified equations. We conclude

AbstractNon-Gaussianity is an important resource for quantum information processing with continuous variables. We introduce a measure of the non-Gaussianity of bosonic field states based on the Hellinger distance and present its basic features. This measure has some natural properties and is easy to compute. We illustrate this measure with typical examples of bosonic field states and compare it with

AbstractWe consider the one-dimensional Schrödinger operator with a semiclassical small parameter \(h\). We show that the “global” asymptotic form of its bound states in terms of the Airy function “works” not only for excited states \(n\sim1/h\) but also for semi-excited states \(n\sim1/h^\alpha\), \(\alpha>0\), and, moreover, \(n\) starts at \(n=2\) or even \(n=1\) in examples. We also prove that

AbstractWe study a “hard-core” model on a Cayley tree. In the case of a normal divisor of index 4, we show the uniqueness of weakly periodic Gibbs measures under certain conditions on the parameters. Moreover, we prove that there exist weakly periodic (nonperiodic) Gibbs measures different from those previously known.

AbstractWe propose a method for constructing a perturbation theory with a finite radius of convergence for a rather wide class of quantum field models traditionally used to describe critical and near-critical behavior in problems in statistical physics. For the proposed convergent series, we use an instanton analysis to find the radius of convergence and also indicate a strategy for calculating their

AbstractWe consider the problem of deformation quantization of presymplectic manifolds in the framework of the Fedosov method. A class of special presymplectic manifolds is distinguished for which such a quantization can always be constructed. We show that in the general case, the obstructions to quantization can be identified with some special elements of the third cohomology group of a differential

AbstractIf an initial state \(|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle \) and a final state \(|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle \) are given, then there exist many Hamiltonians under whose action \(|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle \) evolves into \(|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle \). In this case, the problem of the transition of \(|\psi_{ \s

AbstractWe discuss the extent to which the often assumed independence of the phase of the elastic scattering amplitude from momentum transfer in the domain of only small \(t\) constrains the dependence of the phase on \(t\) in general. Based on analyticity, we prove that if the scattering amplitude phase is independent of the transferred momentum in the domain of its small values in strong couplings

AbstractWe establish a relation between linear second-order difference equations corresponding to Chebyshev polynomials and Catalan numbers. The latter are the limit coefficients of a converging series of rational functions corresponding to the Riccati equation. As the main application, we show a relation between the polynomials \( \varphi _n(\mu)\) that are solutions of the problem of commutation

AbstractThe mean magnetic field equation describes the process of generating a magnetic field on a large scale as a result the EMF arising on a small scale. We consider the case where the large-scale magnetic field is also random and determine the density function of magnetic helicity. This function is invariant under gauge transformations of the magnetic vector potential. We study the equation for

AbstractWe describe the process of constructing a positive solution of the homogeneous Wiener–Hopf integral equation in an octant in a special \((\)conservative\()\) case. Applying the obtained general results to the homogeneous stationary Peierls equation allows studying the behavior of the solutions of this equation for large argument values. These problems are particularly interesting in the theory

AbstractWe analytically solve a boundary value problem for the behavior \((\)oscillation\()\) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in a half-space with mirror boundary conditions. We apply the Vlasov–Boltzmann kinetic equation with a collision integral of the Bhatnagar–Gross–Krook type and a Poisson equation for the electric field. We obtain the electron

AbstractWe introduce a functional version of the Kato one-parameter regularization for constructing a dynamical semigroup generator of a relative bound-one perturbation. As an example of an application, we consider a regularization based on a boson-number cutoff.

AbstractWe prove the existence of a thermal rectification mechanism in a harmonic model with a temperature-dependent effective potential. In contrast to much earlier work where it was shown for this model that rectification occurs in short chains of up to six sites, we analytically prove that this phenomenon occurs in a material with graded mass distribution for any size of the chain and is independent

AbstractWe establish Liouville correspondences for the integrable two-component Camassa–Holm hierarchy, the two-component Novikov \((\)Geng–Xue\()\) hierarchy, and the two-component dual dispersive water wave hierarchy using the related Liouville transformations. This extends previous results for scalar Camassa–Holm and KdV hierarchies and Novikov and Sawada–Kotera hierarchies to the multicomponent

AbstractUsing a self-consistent field, we construct a complete theory of the macroscopic description of cosmological evolution, including a subsystem of linear equations for the evolution of perturbations and nonlinear macroscopic Einstein equations and a scalar field. We present example solutions of this system showing the principal difference between cosmological models of the early universe constructed

We prove that solutions of the Cauchy problem for the nonlinear Schrödinger equation with certain initial data collapse in a finite time, whose exact value we estimate from above. We obtain an estimate from below for the solution collapse rate in certain norms.

We present results of a group analysis of the multidimensional Boltzmann and Vlasov equations. For the Boltzmann equation, we obtain the equivalence group and classifying relations for the symmetry group and study these relations in the case where external forces are absent. We discover a scale paradox: we show that for any collision integral, an equation that is invariant with respect to the shift

We derive time evolution equations for the scattering data for the matrix Zakharov-Shabat system with a potential that is the solution of the matrix modified Korteweg-de Vries equation with an integral-type source.

We investigate higher-order nonlinear Schrödinger-Maxwell-Bloch equations using the Riemann-Hilbert method. We perform a spectral analysis of the Lax pair and construct a Riemann-Hilbert problem according to the spectral analysis. As a result, we obtain three types of multisoliton solutions. Based on the analytic solution and with a choice of corresponding parameter values, we obtain solutions of the

Based on fundamental principles of quantum mechanics, we present a method for a rigorous analytic construction of the spectra of the one- and two-photon Jaynes-Cummings models in a Kerr medium. To obtain an idea of the method, we consider a first generalized Jaynes-Cummings model with a real, linear superpotential using techniques of supersymmetric quantum mechanics. The Hamiltonian of this model is

We consider the Bäcklund transformation for the Degasperis-Procesi (DP) equation. Using the reciprocal transformation and the associated DP equation, we construct the Bäcklund transformation for the DP equation involving both dependent and independent variables. We also obtain the corresponding nonlinear superposition, which we use together with the Bäcklund transformation to derive some soliton solutions

As is known, a nonnegative-definite Hamiltonian H that has a tridiagonal matrix representation in a basis set allows defining forward (and backward) shift operators that can be used to determine the matrix representation of the supersymmetric partner Hamiltonian H(+) in the same basis. We show that if the Hamiltonian is also shape-invariant, then the matrix elements of the Hamiltonian are related such

We derive a general formula giving a representation of the partition function of the one-dimensional Ising model of a system of N particles in the form of an explicitly defined functional of the spectral invariants of finite submatrices of a certain infinite Toeplitz matrix. We obtain an asymptotic representation of the partition function for large N, which can be a base for explicitly calculating

The problem of quantizing the space Ωd of smooth loops taking values in the d-dimensional vector space can be solved in the framework of the standard Dirac approach. But a natural symplectic form on Ωd can be extended to the Hilbert completion of Ωd coinciding with the Sobolev space Vd:= H01/2 (\(\mathbb{S}^1\), ℝd) of half-differentiable loops with values in ℝd. We regard Vd as the phase space of

We extend the biquaternionic Dirac equation to include interactions with a background bosonic Geld. The obtained biquaternionic Dirac equation yields Maxwell-like equations that hold for both a matter Geld and an electromagnetic Geld. We establish that the electric Geld is perpendicular to the matter magnetic Geld and the magnetic Geld is perpendicular to the matter inertial Geld. We show that the

We present a further development of the algebraic aspect of gauge theories initiated in our previous papers.

We study the impact of surface defects on a condensate of electron pairs in a quantum wire. Based on previous results, we formulate a simple mathematical model accounting for such surface effects. For a system of noninteracting pairs, we prove the destruction of the condensate in the bulk. Finally, taking repulsive interactions between the pairs into account, we show that the condensate is recovered

For a wide class of two-particle Schrödinger operators H(k) = H0(k) + V, k ∈ \(\mathbb{T}^d\), corresponding to a two-fermion system on a d-dimensional cubic integer lattice (d ≥ 1), we prove that for any value k ∈ \(\mathbb{T}^d\) of the quasimomentum, the discrete spectrum of H(k) below the lower threshold of the essential spectrum is a nonempty set if the following two conditions are satisfied.

We use the method of the inverse spectral problem to integrate the nonlinear Korteweg–de Vries equation with an additional term in the class of periodic functions.

We calculate exact two-dimensional analytic expressions for the electric and magnetic fields and their potentials generated by a linear beam of relativistic charged particles between infinite ideally conducting planes and ferromagnetic poles. We use the method of summing an infinite sequence of fields of linear image charges and image currents to obtain solutions and also calculate the density distribution

We develop a well-posed multiparticle Coulomb scattering problem based on asymptotic solutions of a Coulomb multichannel scattering problem constructed in the adiabatic representation. A key feature of the studied problem is that the nonadiabatic channel coupling matrix is nontrivial at large internuclear distances. We consider the scattering problem in the case with an arbitrary momentum value and

We propose a new approach for calculating multisoliton solutions of the Degasperis–Procesi equation and its shortwave limit by combining a reciprocal transformation with the Darboux transformation of the negative flow of the Kaup–Kupershmidt hierarchy. In particular, different specifications of the soliton parameters lead to two different types of soliton solutions of the Degasperis–Procesi equation

We study the two-dimensional scattering of a quantum particle by the force center acting on this particle via a superposition of a Coulomb potential and a power-law potential V0r−β with the exponent β ∈ (1,2). We find the explicit low-energy asymptotic forms of all partial phases and scattering amplitudes.

The Kolmogorov spectrum of waves on water is a result of cascading energy via four-wave interactions. For this spectrum in the isotropic case, we introduce basic small perturbations and calculate the decrement of their damping depending on the frequency. We confirm the stability of the Kolmogorov spectrum. The calculation results are applicable for analyzing the stability of numerical methods for solving

We consider the modernized Camassa—Holm equation with periodic boundary conditions. The quadratic nonlinearities in this equation differ substantially from the nonlinearities in the classical Camassa—Holm equation but have all its main properties in a certain sense. We study the so-called nonregular solutions, i.e., those that are rapidly oscillating in the spatial variable. We investigate the problem

We study the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables. By integrability, we mean the presence of reductions of a chain to a system of hyperbolic equations of an arbitrarily high order that are integrable in the Darboux sense. Darboux integrability admits a remarkable algebraic interpretation: the Lie—Rinehart algebras related

Using the exact Hasselmann equation, we study wind-driven deep-water ocean waves in a strait with the wind directed orthogonally to the shore. The strait has “dissipative” shores with no reflection from the shorelines. We show that the evolution of wave turbulence can be divided into two different regimes in time. During the first regime, waves propagate with the wind, and the wind-driven sea can be

Based on numerical simulation of the nonstationary working of a CW superradiant laser with a low-Q combined Fabry-Perot cavity with distributed feedback of waves, we describe a parametric mechanism of self-mode-locking caused by beats of two superradiant modes with a period half that of the cavity round-trip period and supporting the formation of a soliton of the electromagnetic field inside the cavity

We present the concept of an adiabatic limit of Ginzburg—Landau dynamical equations on ℝ1+2 and Seiberg—Witten equations on four-dimensional symplectic manifolds. We show that the Seiberg—Witten equations can be regarded as a complex version of the Ginzburg—Landau equations.

We consider a boundary-value problem for a quasilinear partial differential equation describing tube oscillations under the action of a fluid flow. We show that the well-known Landau—Hopf scenario of transition to turbulence is realized in the considered evolution boundary-value problem with a suitable choice of the governing parameter. To study the problem, we use the theory of infinite-dimensional

For distributed evolutionary dynamical systems of the “reaction-diffusion” and “reaction-diffusion-advection” types, we analyze the behavior of invariant numerical characteristics of the attractor as the diffusion coefficients decrease. We consider the phenomenon of multimode diffusion chaos, one of whose signatures is an increase in the Lyapunov dimensions of the attractor. For several examples, we

We study the dynamics of the generalized Mackey—Glass equation with two delays using the large-parameter method. After a special exponential replacement of variables, a singularly perturbed equation is obtained, for which we construct a meaningful limit object, a differential—difference relay equation with two delays. We prove that the relay equation has a simple periodic solution with one interval

We classify and analyze transformations of three-dimensional dissipative tangle-solitons, i.e., solitons with closed and nonclosed vortex lines, under smooth variations of the system parameters. An example of such a system is a laser medium with fast saturable absorption. We find several scenarios of both reversible and irreversible transformations and discuss the role of the dissipative property of

We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the

We consider a system of nonlinear autonomous differential equations that describe the orientation of the antiferromagnetic vector in a multiferroic film and find the conditions for the existence of spatially modulated structures of the cycloid type. We investigate the stability of such structures under spatial perturbations.

We show that the Euler equations describing the unsteady potential flow of a two-dimensional deep fluid with a free surface in the absence of gravity and surface tension can be integrated exactly under a special choice of boundary conditions at infinity. We assume that the fluid surface at infinity is unperturbed, while the velocity increase is proportional to distance and inversely proportional to