Theoretical and Mathematical Physics最新文献

We asymptotically obtain a generalized Schrödinger equation for nonlinear deformation waves in a metamaterial. It turns out to be analogous to the Sasa–Satsuma equation derived for optical waves. We study distinctions in the solution in the form of localized deformation waves related to the generalization of the Schrödinger equation.

As a result of the linearization of nonlinear equations for displacements in a nonlinear model of elastically conductive micropolar medium in a magnetic field on the background of a soliton solution describing subsonic solitary waves, we obtain an inhomogeneous scalar linear equation. This equation leads to a generalized spectral problem. To establish the instability of the mentioned solitary waves, the existence of an unstable eigenvalue (with a positive real part) must be verified. The corresponding proof is carried out by constructing the Evans function that depends only on the spectral parameter. This function is analytic in the right complex half-plane, and its zeros coincide with the unstable eigenvalues. It is proved that the Evans function tends to unity at infinity. This property of the Evans function, for some of its local properties in a neighborhood of the origin, allows us to conclude that it has zeros on the positive real semi-axis and therefore the subsonic solitary wave is unstable.

We propose a procedure for constructing exact solutions of equations of nonlinear mathematical physics based on the application of the Shanks extrapolation method to a segment of a perturbation series in powers of exponents that are solutions of a sequence of linear problems. We assume that a sequence of partial sums of the power series belongs to the Shanks transformation kernel. In the Shanks method, the initial value of the order of the linear combination is chosen to be one greater than the order of the pole of the solution to the original equation. The efficiency of the method is demonstrated in the construction of exact localized solutions of a nonlinear heterogeneous ordinary differential equation, the generalized Tzitzéica equation, as well as its difference and differential–difference analogues.

We consider solutions of a system of magnetoelasticity equations. As initial data for these solutions, we use data of the smoothed step type (the problem of discontinuity decay). Among these solutions, there are solutions with purely nondissipative structures of the soliton type and structures with the radiated wave and the internal dissipative discontinuities of derivatives. We develop techniques for studying discontinuities in solutions of equations with dispersion and finite of wave propagation velocity. We analyze and justify the existence of such structures by studying equations of traveling waves. We reveal the presence of sequences of weak discontinuities in structures with the radiated wave. We also study a dissipative structure of the shock-wave type. We consider conditions for discontinuities and their evolutionary properties. We establish that when studying the discontinuities in the solutions of dispersion equations, the limiting velocities of short waves play the same role as the characteristic velocities for hyperbolic equations.

For each of the forty-eight exceptional algebraic solutions (u(x)) of the sixth Painlevé equation, we build the algebraic curve (P(u,x)=0) of a degree conjectured to be minimal, and then we give an optimal parametric representation of it. This degree is equal to the number of branches, except for fifteen solutions.

We consider issues related to the formulation of problems of describing the dynamics of linear internal gravity waves in stratified media with horizontal shear flows under critical wave generation modes. In a plane setting, we discuss new model physical formulations of the problems where critical modes may occur. For arbitrary distributions of the buoyancy frequency and shear flows satisfying the Miles–Howard conditions and natural regularity conditions, we study analytical properties of solutions of the main spectral problem of the internal gravity waves equation with shear flows under critical wave generation modes for the cases of simlpe and multiple eigenvalues.

The quantum calculus with two bases, represented by powers of the golden and silver ratios, relates the Fibonacci divisor derivative with Binet formula for the Fibonacci divisor number operator, acting in the Fock space of quantum states. It provides a tool to study the hierarchy of golden oscillators with energy spectrum in the form of Fibonacci divisor numbers. We generalize this model to the supersymmetric number operator and corresponding Binet formula for the supersymmetric Fibonacci divisor number operator. The operator determines Hamiltonian of the hierarchy of supersymmetric golden oscillators, acting in fermion–boson Hilbert space and belonging to (N=2) supersymmetric algebra. The eigenstates of the super Fibonacci divisor number operator are double degenerate and can be characterized by a point on the super-Bloch sphere. By introducing the supersymmetric Fibonacci divisor annihilation operator, we construct the hierarchy of supersymmetric coherent states as eigenstates of this operator. The entanglement of fermions with bosons in these states is calculated by the concurrence, represented as the Gram determinant and expressed in terms of the hierarchy of golden exponential functions. We show that the reference states and the corresponding von Neumann entropy measuring the fermion–boson entanglement are characterized completely by powers of the golden ratio. We give a geometrical classification of entangled states by the Frobenius ball and interpret the concurrence as the double area of a parallelogram in a Hilbert space.

We consider an extended version of the second Painlevé equation ((mathrm P_{mathrm{II}})), which appears as the simplest member of a recently-derived extended second Painlevé hierarchy. For this third-order system we consider the application of the Ablowitz–Ramani–Segur algorithm, use its auto-Bäcklund transformations ( auto-BTs) to construct sequences of rational solutions and solutions defined in terms of Bessel functions, the latter constituting the analogues for the extended (mathrm P_{mathrm{II}}) of the well-known Airy function solutions of (mathrm P_{mathrm{II}}). In addition, we present two new Bäcklund transformations, which extend the Schwarzian (mathrm P_{mathrm{II}}) equation due to Weiss and an auto-BT due to Gambier. Finally, we use the auto-BTs of extended (mathrm P_{mathrm{II}}) also to derive a new third-order discrete system.

We find implicit quiescent solitons in optical metamaterials with nonlinear chromatic dispersion and several forms of self-phase modulation structures. The temporal evolution is however assumed to be linear. Some of the self-phase modulation structures yield results with quadratures. In each case, the governing model is integrated by applying Lie symmetry.

We consider a boundary value problem with a time-periodic condition for an equation of “reaction–advection–diffusion” type with weak smooth advection and with reaction discontinuous in the spatial coordinate. We construct the asymptotics, prove the existence, and investigate the stability of periodic solutions with the constructed asymptotics and with a weak internal layer formed near the discontinuity point. To construct the asymptotics, we use the Vasil’eva method; to justify the existence of the solution, the asymptotic method of differential inequalities; and to study stability, the method of contracting barriers. We show that such a solution, as a solution of the corresponding initial-boundary value problem, is asymptotically Lyapunov stable. We determine the stability domain of a finite (not asymptotically small) width for such a solution and prove that the solution of the periodic problem is unique in this domain.