Abstract We study the problem of the absence of global periodic solutions of a nonlinear Schrödinger-type evolution equation with a damped linear term. We prove that in the case where the damping factor is negative, the problem has no global periodic solutions for any initial data, and in the case of a negative damping factor, this is also true for large enough values of the initial data.

Abstract We consider three one-dimensional superconducting structures: 1) the one with \(p\)-wave superconductivity; 2) the main experimental model of a nanowire with \(s\)-wave superconductivity generated by the bulk superconductor due to the proximity effect in an external magnetic field and Rashba spin–orbit interaction; 3) the boundary of a two-dimensional topological insulator with an \(s\)-wave

Abstract We construct a Hamiltonian field theory model that combines hydrodynamics and thermodynamics. In this model, the continuity equation and the Euler equation for a potential flow are used as the Hamilton equations; and equations of state (or the Gibbs equations, which are equivalent to them ), as equations of second-class constraints. The Hamiltonian function density reduced to the surface of

Abstract We discuss the stationary potential equations as illustrative examples to explain how to construct integrable symplectic maps via Bäcklund transformations. We first give a terse survey of Bäcklund transformations of the potential KdV equation and the potential fifth-order KdV equation. Then, using Jacobi–Ostrogradsky coordinates, we obtain canonical Hamiltonian forms of the stationary potential

Abstract There is an extensive literature on the dynamic law of large numbers for systems of quantum particles, that is, on the derivation of an equation describing the limiting individual behavior of particles in a large ensemble of identical interacting particles. The resulting equations are generally referred to as nonlinear Schrödinger equations or Hartree equations, or Gross–Pitaevskii equations

Abstract We consider two different subjects: the \(q\)-deformed universal characters \(\widetilde S_{[\lambda,\mu]}(t,\hat t;x,\hat x)\) and the \(q\)-deformed universal character hierarchy. The former are an extension of \(q\)-deformed Schur polynomials, and the latter can be regarded as a generalization of the \(q\)-deformed KP hierarchy. We investigate solutions of the \(q\)-deformed universal character

Abstract We consider nonperturbative vacuum polarization effects in the supercritical region for a planar Dirac–Coulomb system with a supercritical extended axially symmetric Coulomb source with a charge \(Z > Z_{{\rm cr},1}\) and radius \(R_0\) in the magnetic field with an axial-vector potential. We study the behavior of the vacuum charge and vacuum current densities, \(\rho_{\mathrm{VP}}(\vec r\

Abstract We give a general and weak sufficient condition that is very close to a necessary and sufficient condition for the existence of a sequence of solutions converging to zero for the partial differential equations known as the \(p\)-Laplacian Hamiltonian systems. An application is also given to illustrate our main theoretical result.

Abstract Although variance, as one of the most fundamental and ubiquitous quantities in quantifying uncertainty, has been widely used in both classical and quantum physics, there are still new applications awaiting exploration. In this work, by interchanging the roles of the state variable and the observable variable, i.e., by formally regarding any state as an observable (which is rational because

Abstract This is a short review of the Kadomtsev–Petviashvili hierarchies of types \(\mathrm{B}\) and \(\mathrm{C}\). The main objects are the \(L\) operator, the wave operator, the auxiliary linear problems for the wave function, the bilinear identity for the wave function, and the tau function. All of them are discussed in the paper. Connections with the usual (type-\(\mathrm{A}\)) Kadomtsev–Petviashvili

Abstract Integrals of motion are constructed from noncommutative (NC ) Kepler dynamics, generating \(\mathrm{SO}(3)\), \(\mathrm{SO}(4)\), and \(\mathrm{SO}(1,3)\) dynamical symmetry groups. The Hamiltonian vector field is derived in action–angle coordinates, and the existence of a hierarchy of bi-Hamiltonian structures is highlighted. Then, a family of Nijenhuis recursion operators is computed and

Abstract We use the group-based discrete moving frame method to study invariant evolutions in a \(n\)-dimensional centro-affine space. We derive the induced integrable equations for invariants, which can be transformed to local and nonlocal multi-component Toda lattices under a Miura transformation, and thus establish their geometric realizations in centro-affine space.

Abstract We propose a model of a ring circuit of \(m\) generators that is a relay analog of a circuit of Mackey–Glass generators. In this model, each of the generators is described by the limit Mackey–Glass equation. For this relay system, we prove the existence of a periodic solution of discrete traveling wave type, i.e., a solution all of whose \(m\) components (describing the \(m\) generators )

Abstract We study a two-dimensional singularly perturbed system with delay, which is a simplification of models used in laser physics. We analyze several cases of a small parameter multiplying the derivative in the first equation and investigate the behavior of solutions in a neighborhood of a stationary point when the system parameters pass through bifurcation values. Methods for local asymptotic

Abstract We propose a method for analyzing the Cauchy problem for a wide class of equations with power-like nonlinearities. The method is based on the Fourier transformation, which allows reducing the original equation to an integro-differential one. We prove the existence of solutions.

Abstract We study the existence and uniqueness of the Kadomtsev–Petviashvili (KP ) hierarchy solutions in the algebra \(\mathcal FCl(S^1,\mathbb K^n)\) of formal classical pseudodifferential operators. The classical algebra \(\Psi DO(S^1,\mathbb K^n)\), where the KP hierarchy is well known, appears as a subalgebra of \(\mathcal FCl(S^1,\mathbb K^n)\). We investigate algebraic properties of \(\mathcal

Abstract Dynamic systems acting on the plane and possessing the Wada property have been observed. There exist only two topological types, symmetric and antisymmetric, of dissipative dynamic systems with the nonwandering continuum being a common boundary of three regions. An antisymmetric dynamic system with the nonwandering continuum can be transformed into a dynamic system with an invariant vortex

Abstract We consider the Cahn–Hilliard equation in the case where its solution depends on two spatial variables, with homogeneous Dirichlet and Neumann boundary conditions, and also periodic boundary conditions. For these three boundary value problems, we study the problem of local bifurcations arising when changing stability by spatially homogeneous equilibrium states. We show that the nature of bifurcations

Abstract We study a difference–differential model of an optoelectronic oscillator that is a modification of the Ikeda equation with delay. We analyze the stability of the zero equilibrium state. We note that the number of roots of the characteristic equation of the linearized problem with the real part that is close to zero increases without bound for order parameter values tending to bifurcational

Abstract A logistic equation with a delay depending on the state in the past is considered. We study the question of the existence of nonlocal relaxation periodic solutions of this equation for large values of the parameter. The characteristics (for example, period and extreme values) of solutions that we find are compared with similar characteristics of solutions of other modifications of this equation

Abstract An overview of affine fractal interpolation functions using a suitable iterated function system is presented. Furthermore, a brief and coarse discussion on the theory of affine fractal interpolation functions in 2D and their recent developments including some of the research done by the authors is provided. Moreover, the desired range of the contractivity factors of an affine fractal interpolation

Abstract We show that any extension of an Abelian group corresponds to a solution of the parametric Yang–Baxter equation. This statement is a generalization of the well-known construction of a braided set in terms of group structure to the case of group extensions. We also show that this construction in the case of a semidirect product is a specialization of a more general construction using principal

Abstract We study the problem of the existence and asymptotic stability of a stationary solution of an initial boundary value problem for the reaction–diffusion–advection equation assuming that the reaction and advection terms are comparable in size and have a jump along a smooth curve located inside the studied domain. The problem solution has a large gradient in a neighborhood of this curve. We prove

Abstract We construct a generalization of the Tutte polynomial for vertex-weighted graphs for which the coefficients of the “deletion–contraction” relation depend nontrivially on the vertex weights. We show that the corresponding relation on the coefficients coincides with the two-cocycle relation in the group cohomology. We obtain a representation of a new invariant by summing over subgraphs and establish

Abstract We study certain extensions of the Adler map on Grassmann algebras \(\Gamma(n)\) of order \(n\). We consider a known Grassmann-extended Adler map and under the assumption that \(n=1\), obtain a commutative extension of the Adler map in six dimensions. We show that the map satisfies the Yang–Baxter equation, admits three invariants, and is Liouville integrable. We solve the map explicitly by

Abstract We define quantum determinants in quantum matrix algebras related to pairs of compatible braidings. We establish relations between these determinants and the so-called column and row determinants, which are often used in the theory of integrable systems. We also generalize the quantum integrable spin systems using generalized Yangians related to pairs of compatible braidings. We demonstrate

Abstract We explain the details of the algebraic constructions on nontrivial examples of the mKdV equations related to the \(A_5^{(1)}\) and \(A_5^{(2)}\) Kac–Moody algebras. Several types of recursion operators appear naturally in formulating the equations and their Hamiltonian structures. We next introduce the resolvent of the Lax operator and demonstrate that it generates the hierarchy of the Lax

Abstract We use the Painlevé–Kovalevskaya test to find three matrix versions of the Painlevé II equation. We interpret all these equations as group-invariant reductions of integrable matrix evolution equations, which allows constructing isomonodromic Lax pairs for them.

Abstract We study a generalized Hénon map in two-dimensional space. We find a region of the phase space where the nonwandering set exists, specify parameter values for which this nonwandering set is hyperbolic, and prove that our map when restricted to a specific invariant subset is topologically conjugate to the Bernoulli three-shift. Coupling two such maps, as a result, we obtain a map in four-dimensional

Abstract We consider the problem of three-dimensional motion of a passively gravitating point in the potential created by a homogeneous thin fixed ring and a massive point located in the center of the ring. The motion of a passively gravitating point admits two first integrals. We first consider the integrable case of an invariant motion in the equatorial plane and then consider the general case of

Abstract We consider chains of van der Pol equations closed into a ring and chains of systems of two first-order van der Pol equations. We assume that the couplings are homogeneous and the number of chain elements is sufficiently large. We naturally realize a transition to functions depending continuously on the spatial variable. As \(t\to\infty\), we study the behavior of all solutions of such chains

Abstract We construct an asymptotic expansion in a small parameter of a boundary layer solution of the boundary value problem for a system of two ordinary differential equations, one of which is a second-order equation and the other is a first-order equation with a small parameter at the derivatives in both equations. Such a system arises in chemical kinetics when modeling the stationary process in

Abstract We present the probability distribution function of the nodal domain number for the right-isosceles triangle billiard. This is the first explicit result for a nonseparable plane polygonal billiard. The distribution function has a peak at \(2/\pi\) (the same as for integrable billiards) and also at \(2\sqrt{2}/\pi\). The only exact results previously known in this direction correspond to integrable

Abstract We study a complete cosmological model based on an asymmetric scalar doublet represented by the classical and phantom scalar Higgs fields. Moreover, we remove the assumption that the expansion rate of the Universe is nonnegative, which contradicts the complete system of Einstein’s equations in several cases. We formulate a closed system of dynamical equations describing the evolution of the

Abstract Using the Hamiltonian structures and tau symmetries, we construct a multicomponent fractional Volterra hierarchy (MFVH) as an example of a generalized integrable system. We study the Hirota bilinear equation and the Virasoro symmetry for this hierarchy. As a reduction of the MFVH from a commutative subalgebra, we construct the fractional \(Z_N\)-Volterra hierarchy and its Virasoro symmetry

Abstract We use the Riemann–Hilbert (RH) method to study the Kundu-type nonlinear Schrödinger (Kundu–NLS) equation with a zero boundary condition in the case where the scattering coefficient has \(N\) distinct arbitrary-order poles. We perform a spectral analysis of the Lax pair and consider the asymptotic property, symmetry, and analyticity of the Jost solution. Based on these results, we formulate

Abstract We study a semi-infinite metric path graph and construct the long-time asymptotic logarithm of the number of possible endpoints of a random walk.

Abstract We find a solution of the black hole type for a five-dimensional fully anisotropic holographic model for heavy quarks. The model is described by the Einstein action with a dilaton field and three Maxwell tensors. The additional Maxwell term is related to an external magnetic field. We consider the influence of the external magnetic field on the solution for a five-dimensional black hole.

Abstract We study the two-dimensional scattering of a quantum particle in the field of a central long-range potential decaying in the limit of a large distance \(r\) as a power-law function \(r^{-\beta}\) with \(\beta>2\). We find the explicit low-energy asymptotic forms of all partial scattering phases of such a particle.

Abstract We consider the inverse problems of simultaneously determining two unknowns: the wave propagation velocity and the memory of a layered medium. To find them, we use the data of two observed fluctuations on the domain boundary. Estimates of the solution stability and uniqueness theorems for the considered problems are the main results.

Abstract We construct an exactly solvable model of a linear harmonic oscillator with a coordinate-dependent mass in a uniform gravitational field. This model is placed in an infinitely deep potential well with the width \(2a\) and corresponds to the exact solution of the angular part of the Schrödinger equation with one of the Hautot potentials. The wave functions of the oscillator model are expressed

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 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.

We study systematically a matrix Riemann-Hilbert problem for the modified Landau-Lifshitz (mLL) equation with nonzero boundary conditions at infinity. Unlike the zero boundary conditions case, there occur double-valued functions during the process of the direct scattering. In order to establish the Riemann-Hilbert (RH) problem, it is necessary to make appropriate modification, that is, to introduce

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

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

We consider dimer model on a hexagonal lattice. This model can be seen as a "pile of cubes in the box". The energy of configuration is given by the volume of the pile and the partition function is computed by the classical MacMahon formula or, more formally, by the determinant of Kasteleyn matrix. We use the expression for the partition function to derive the scaling behavior of free energy in the

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 coefficients

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 the behavior

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.

Using the adjoint representations of Lie algebras, we classify all Jacobi structures on real two- and three-dimensional Lie groups. Also, we study Jacobi-Lie systems on these real low-dimensional Lie groups. Our results are illustrated through examples of Jacobi-Lie Hamiltonian systems on some real two- and three-dimensional Lie groups.

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. We consider

In this article, we consider the phenomenon of complete coincidence of the key properties of pairs 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 as a hypersurface in the orbifold of another weighted projective space. The two