Abstract We construct and study Darboux transformations for the \((2{+}1)\)-dimensional Camassa–Holm system. We apply a reciprocal transformation that relates the \((2{+}1)\)-dimensional Camassa–Holm system and the linear system associated with the modified Kadomtsev–Petviashvili hierarchy. Using three Darboux transformation operators, we obtain three types of solutions for the \((2{+}1)\)-dimensional

Abstract We study an unbounded \(2{\times}2\) operator matrix \( \mathcal{A} \) in the direct product of two Hilbert spaces. We obtain asymptotic formulas for the number of eigenvalues of \( \mathcal{A} \). We consider a \(2{\times}2\) operator matrix \( \mathcal{A} _\mu\), where \(\mu>0\) is the coupling constant, associated with the Hamiltonian of a system with at most three particles on the lattice

Abstract Using the Noether symmetry approach, we investigate \(f( \mathcal{R} , \varphi ,\chi)\) theories of gravity, where \( \mathcal{R} \) is the scalar curvature, \( \varphi \) is the scalar field, and \(\chi\) is the kinetic term of \( \varphi \). Based on the Lagrangian for \(f( \mathcal{R} , \varphi ,\chi)\) gravity, we obtain the determining equations. We consider \(f( \mathcal{R} , \varphi

Abstract We consider the eigenvalue problem for the two-dimensional Hartree operator with a small nonlinearity coefficient. We find the asymptotic eigenvalues and asymptotic eigenfunctions near a local maximum of the eigenvalues in spectral clusters formed near the eigenvalues of the unperturbed operator.

Abstract Using the Bogoliubov–de Gennes Hamiltonian, we analytically study two models with superconducting order, the p-wave model with an impurity potential and the s-wave nanowire model with superconductivity induced by the proximity effect with an impurity potential in a Zeeman field with a spin–orbit interaction. Using the Dyson equation, we study conditions for the emergence of Andreev bound states

Abstract We present two ways to obtain precise expressions for the commuting Hamiltonians of the integrable system regarded as a fermionic limit of the quantum Calogero–Sutherland system as the number of particles tends to infinity with some special values of the coupling constant \(\beta\). The construction is realized in the Fock space.

Abstract We prove Gaussian fluctuations for pair-counting statistics of the form \(\Sigma_{1\le i\ne j\le N}f(\theta_i-\theta_j)\) for the circular unitary ensemble of random matrices in the large-\(N\) limit under the condition that the variance increases slowly as \(N\) increases.

Abstract We study a matrix Riemann–Hilbert (RH) problem for the modified Landau–Lifshitz (mLL) equation with nonzero boundary conditions at infinity. In contrast to the case of zero boundary conditions, multivalued functions arise during direct scattering. To formulate the RH problem, we introduce an affine transformation converting the Riemann surface into the complex plane. In the direct scattering

Abstract We explicitly express the fundamental solution of the stationary two-dimensional massless Dirac equation with a constant electric field in terms of Fourier transforms of parabolic cylinder functions. This solution describes the flux of quasiparticles in graphene emitted by a pointlike source of electrons that are partially converted into holes (antiparticles). Using our explicit formula, we

Abstract The approach based on commutator identities for elements of associative algebras was previously effectively used to investigate \((2{+}1)\)-dimensional integrable systems. We develop this approach to investigate integrable hierarchies and their relations.

Erratum to https://doi.org/10.1134/S0040577920090068

Abstract This paper is devoted to the proper-time method and describes a model case that reflects the subtleties of constructing the heat kernel, is easily extended to more general cases (curved space, manifold with a boundary), and contains two interrelated parts: an asymptotic expansion and a path integral representation. We discuss the significance of gauge conditions and the role of ordered exponentials

Abstract We consider the dimer model on a hexagonal lattice. This model can be represented as a “pile of cubes in a box.” The energy of a configuration is given by the volume of the pile. The partition function is computed by the classical MacMahon formula or as the determinant of the Kasteleyn matrix. We use the MacMahon formula to derive the scaling behavior of free energy in the limit as the lattice

Abstract Let \(R\) be a commutative complex algebra and \( \partial \) be a \( \mathbb{C} \)-linear derivation of \(R\) such that all powers of \( \partial \) are \(R\)-linearly independent. Let \(R[ \partial ]\) be the algebra of differential operators in \( \partial \) with coefficients in \(R\) and \( P{\kern-1.5pt}sd \) be its extension by the pseudodifferential operators in \( \partial \) with

Abstract We consider interactions of scalar particles, photons, and fermions in Schwarzschild, Reissner–Nordström, Kerr, and Kerr–Newman gravitational and electromagnetic fields with a zero and nonzero cosmological constant. We also consider interactions of scalar particles, photons, and fermions with nonextremal rotating charged black holes in a minimal five-dimensional gauge supergravity. We analyze

Abstract We use the binary Darboux transformation to obtain exact multisoliton solutions of the \(U(N)\) generalized Heisenberg magnet model and present the solutions in terms of quasideterminants. In addition, based on using the Poisson bracket algebra, we develop a new canonical approach of the type of the \(r\)-matrix approach for the generalized Heisenberg magnet model.

Abstract Using the adjoint representations of Lie algebras, we classify all Jacobi structures on real two- and three-dimensional Lie groups. We also study Jacobi–Lie systems on these real low-dimensional Lie groups and illustrate our results with examples of Jacobi–Lie Hamiltonian systems on some real two- and three-dimensional Lie groups.

Abstract In the context of modified \(f( \mathcal{G} )\) gravity, we study the appearance of anisotropic compact stellar objects. The space–time geometry is characterized by the metric potentials, and to solve the field equations, we consider their specific form known as the Tolman–Kuchowicz space–time. The obtained solutions are free of singularities and satisfy all requirements for stellar objects

Abstract We consider the phenomenon of the complete coincidence of key properties of Calabi–Yau manifolds realized as hypersurfaces in two different weighted projective spaces. More precisely, the first manifold in such a pair is realized as a hypersurface in a weighted projective space, and the second is realized as a hypersurface in an orbifold of another weighted projective space. The two manifolds

Abstract To construct a phase-equivalent energy-dependent local potential corresponding to a sum of local and nonlocal interactions, we use a simple method for expanding the wave function in a Taylor series up to the third order. We apply the constructed potentials to calculate the scattering phase shifts using the phase equation. The results for scattering in nucleon–nucleon, \(\alpha\)–nucleon, and

Abstract We describe a family of integrable \(GL(NM)\) models generalizing classical spin Ruijsenaars–Schneider systems (the case \(N=1\)) on one hand and relativistic integrable tops on the \(GL(N)\) Lie group (the case \(M=1\)) on the other hand. We obtain the described models using the Lax pair with a spectral parameter and derive the equations of motion. To construct the Lax representation, we

Abstract We construct a self-consistent approximation for calculating the magnetization and Curie temperature in the Ising model on an arbitrary lattice based on using the magnetization and pair correlation of nearest neighbors. We find the Curie temperatures for simple lattices in this approximation and generalize the approximation to the case of lattices diluted in lattice sites or bonds, for which

Abstract Using the Riemann–Hilbert approach, we investigate the two-component generalized Ragnisco–Tu equation. The modified equation is integrable in the sense that a Lax pair exists, but its explicit solutions have some distinctive properties. We show that the explicit one-wave solution is unstable and the two-wave solution preserves only the phase shift but not the wave shape after collision.

Abstract We propose a mathematical apparatus based on binary relations that expands the possibility of traditional analysis applied to problems in mathematical and theoretical physics. We illustrate the general constructions with examples with an algebraic description of Bäcklund transformations of nonlinear systems of partial differential equations and the dynamics of wave packet propagation.

Abstract We consider the Randall–Sundrum model with two branes in which the fields of the Standard Model are localized on the brane with negative tension and the gravitational field and an additional stabilizing scalar Goldberg–Wise field propagate in the space between the branes. We construct a Lagrangian for scalar fluctuations of the gravitational and scalar fields against the background solution

Abstract We recently constructed a series of integrable discrete autonomous equations on a quadratic lattice with a nonstandard structure of higher symmetries. Here, we construct a modified series using discrete nonpoint transformations. We use both noninvertible linearizable transformations and nonpoint transformations that are invertible on the solutions of the discrete equation. As a result, we

Abstract We consider a model with nearest-neighbor interactions and the set \([0,1]\) of spin values on a Bethe lattice (Cayley tree) of arbitrary order. This model depends on a continuous parameter \(\theta\) and is a generalization of known models. For all values of \(\theta\), we give a complete description of the set of translation-invariant Gibbs measures of this model.

Abstract We study the additional symmetries and \(\tau\)-functions of multicomponent Gelfand–Dickey hierarchies, which include classical integrable systems such as the multicomponent Korteweg–de Vries and Boussinesq hierarchies. Using various reductions, we derive B- and C-type multicomponent Gelfand–Dickey hierarchies. We show that not all flows of their additional symmetries survive. We find that

Abstract We obtain a dispersionless integrable system describing a local form of a general three-dimensional Einstein–Weyl geometry with a Euclidean (positive) signature, construct its matrix extension, and show that it leads to the Bogomolny equations for a non-Abelian monopole on an Einstein–Weyl background. We also consider the corresponding dispersionless integrable hierarchy, its matrix extension

Abstract We construct \(q\)-deformed Dirac and chirality operators on the \(q\)-deformed quantum space \(EAdS^2\) using a generalized quantum Ginsparg–Wilson algebra. We show that in the limit \(q\to1\), these operators become the Dirac and chirality operators on the undeformed quantum space \(EAdS^2\).

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.