Energy levels of a two-dimensional anharmonic oscillator: Hill determinant approach

The energy levels of the Schrödinger equation for the potentialx ²+y ²+λ[a _xxx⁴+2a _xyx² y ²+a _yyy⁴] are calculated using Hill determinant approach for several eigenstates and over a wide range of values of perturbation parameters. The obtained numerical results agree with those previously reported by other methods.     

Energy levels of the Schrödinger equation for some rational potentials in two-dimensional space using the inner product technique

The energy levels of a two-dimensional system are calculated for the rational potential,V(x, y; λ, g)=x ²+y ²+λ[x ²/(1+gx ²)+y ²/(1+gy ²)+a _xxx⁴+a _xyx² y ²+a _yyy⁴] using the inner product technique over a wide range of values of the perturbation parameters (g, λ) and for various eigenstates.     

Anisotropy of the Raman scattering measured in the xy plane of a nontwinned YBa2Cu3O7−x single crystal

The temperature dependence of anisotropy of the phonon and electron Raman scattering components was studied by measurements in the xy plane of a nontwinned YBa₂Cu₃O_7?x single crystal. It is shown that the sign of the orthorhombicity parameter γ=(I _yy ?I _xx )/(I _yy +I _xx ) for the full-symmetry phonons (150, 340, and 435 cm^?1 modes) generated by the displacements of ions in the CuO₂ plane is opposite to the sign of this parameter for the phonons generated by the out-of-plane barium and bridging oxygen displacements (120 and 500 cm^?1 modes). In the superconducting state, the γ value decreases in the region of low frequencies, but the frequency renormalization of the 340 cm^?1 mode measured in the xx and yy spectra is the same to within the experimental error.     

Equations for the description of wave scattering by multi-valued random surfaces

We consider a scattering theory for multi-valued rough surfaces which cannot be described by the conventional equation of the type z = ζ(x,y). Both Dirichlet and Neumann problems are analyzed. Starting with Green's theorem we obtain a representation of the scattered field, the surface integral equation, and the extinction theorem for such surfaces. In contrast to conventional theory, these equations contain three random functions x = x(u ₁,u ₂), y = y(u ₁,u ₂), and z = z(u ₁,u ₂), where u ₁ and u ₂ are the parameters describing the surface. We introduce two scattering amplitudes S _± for describing the scattered wave above and below the surface. The extinction theorem, if formulated in terms of S _?, allows us to determine S _? for an arbitrary multi-valued surface and after this it becomes possible to derive a simple integral equation for surface sources. Knowledge of the surface sources allows us to find S ₊ by integration.     

Essentially different distributions of current in the inverse problem for the Grad-Shafranov equation

The paper answers a question debated by physicists for many years. It is proved that, for almost equal gradients of the magnetic flux u at its zero-level curve ∂ω, which is the piecewise smooth boundary of a simply-connected domain ω ⋐ ℝ², the inverse problem for the Grad-Shafranov equation of plasma equilibrium in a tokamak (in the cylindrical approximation) admits essentially different profiles of distributions f _u : ω ∋ (x, y) ↦ f(u(x, y)) = u _xx (x, y) + u _yy (x, y) ⩾ 0 in the class of third-order polynomials f(u) = Σ _m=0³ a _m u ^m .     

A family of integrable evolution equations of third order

We construct a family of integrable equations of the form v_t = f(v; v_x; v_xx; v_xxx) such that f is a transcendental function in v; v_x; v_xx. This family is related to the Krichever-Novikov equation by a differential substitution. Our construction of integrable equations and the corresponding differential substitutions involves geometry of a family of genus two curves and their Jacobians.     

Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation

We prove that the set of solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in two dimensions, (u _t+(u^m+1)_x+u_xxx)_x=u_yy is stable for 0<m<4/3.     

Stability range for a flat graphene sheet subjected to in-plane deformation

The effects of the elastic deformation on the mechanical and physical properties of graphene are a subject of intensive current studies. Nevertheless, the stability range for a flat graphene sheet subjected to in-plane deformation is still unknown. Here, this problem is solved by atomistic simulations. In the three-dimensional space corresponding to the ɛ _xx , ɛ _yy , and ɛ _xy components of the planar strain tensor, the surface bounding the stability range for a flat graphene sheet has been constructed disregarding the thermal vibrations and the effects of boundary conditions. For the points of this surface, force components T _x , T _y , and T _xy have been calculated. It is shown that graphene is structurally stable up to strains on the order of 0.3–0.4, but it is unstable with respect to the shear in the absence of stretching forces. In addition, graphene cannot preserve its flat shape under the effect of a compressive force since it has zero flexural stiffness.     

A higher order lattice BGK model for simulating some nonlinear partial differential equations

In this paper, we consider a one-dimensional nonlinear partial differential equation that has the form ut + αuux + βunux - γuxx + δuxxx = F(u). A higher order lattice Bhatnager-Gross-Krook (BGK) model with an amending-function is proposed. With the Chapman-Enskog expansion, different kinds of nonlinear partial differential equations are recovered correctly from the continuous Boltzmann equation. The numerical results show that this method is very effective.     

Raman scattering in YBa2Cu3O7 single crystals: Anisotropy in normal and superconductivity states

Precise temperature and polarization dependences of Raman spectra have been investigated for fully oxygenated twin-free YBa₂Cu₃O₇ single crystals. We have found a striking superconductivity-induced x−y anisotropy in the temperature behavior of the 340 cm⁻¹ line: the magnitudes of the softening and broadening are quite different in the xx-and yy-polarizations. This anisotropy suggests a contribution of the CuO-chain superconductivity with a pairing symmetry different from that for the CuO₂ plane, or indicates that the superconducting gap amplitudes are different in the k _x and k _y directions. The d+s gap symmetry is the only realistic symmetry in the case of Δ_x≠Δ_y. Fiz. Tverd. Tela (St. Petersburg) 40, 403–412 (March 1998) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Extension of the module of invertible transformations. Classification of integrable systems

We demonstrate that for the systems of equations, which are invariant under a point group or possess conservation laws of the zeroth or first order, a nontrivial extension of the module of invertible transformations is possible. That simplifies greatly a classification of the integrable systems of equations. Here we present an exhaustive list and a classification of the second order systems of the formu _t =u _xx +f(u, v, u _x v _x ), –v _t=v _xx +g(u, v, u _x ,v _x ), which possess the conservation laws of higher order. The reduction group approach allows us to define the Lax type representations for some new equations of our list.     

Bäcklund transformations for diffusion equations

A class of Bäcklund transformations is deduced for the diffusion equation (S(u))_t = (C(u,u_x))_x by taking advantage of the conservative form of the equation. The transformations are used for obtaining new relations connecting various equations of the type u_t = (u^au_x)_x and u_t = u^au_xx.     

Polarization properties of two Gaussian Schell-model beams in the turbulent atmosphere

General expressions are derived for the spectral degree of polarization of a beam generated by the superposition of two Gaussian Schell-model (GSM) beams propagating in a turbulent atmosphere by adopting the beam coherence-polarization matrix and Young's interference theory. We analyzed the distance of two beams, atmospheric turbulence intensity, and the location of the imaging plane affect the degree of polarization by numerical calculation. In particular, when the correlation coefficients of x and y components of the electric field are the same (i.e. δ_xx = δ_yy), the degree of polarization will obtain the same value.     

EPR and optical-absorption studies of Cu2+-doped Zn(NH2(CH2)3NHOH)2Ni(CN)4 single crystals

Electron paramagnetic resonance of Cu²⁺-doped catena-trans-bis(N-(2-hydroxyethyl)-ethylenediamine) zinc(II)-tetra-μ-cyanonicelate(II) [Zn(NH₂(CH₂)₃NHOH)₂Ni(CN)₄] single crystals and powder are examined at room temperature. The spectra show the substitution of the Zn²⁺ ion with the Cu²⁺ ion. The crystal field around the Cu²⁺ ion is nearly axial. There is a single paramagnetic site withg _xx=2.073,g _yy=2.060,g _zz=2.248,A _xx=40.5 G,A _yy=50.8 G,A _zz=172.0 G. The ground-state wave function is an admixture of d _x ²−y ² and d _z ² states. The optical-absorption studies show two bands at 320 nm (31250 cm⁻¹) and 614 nm (16286 cm⁻¹) which confirm the axial symmetry. The crystal field parameters and the wave function are determined.     

The Painlevé property and a partial differential equation with an essential singularity

It is shown that the equation u²_t = 2uu²_x - (1 + u²)u_xx possesses the Painlevé property for partial differential equations as defined by Weiss, Tabor and Carnevale, yet does not satisfy the necessary conditions of the Painlevé conjecture to be completely integrable since it is reducible, via a similarity reduction, to an ordinary differential equation which has a movable essential singularity. It is further shown that in a more general sense, the equation does not possess the Painlevé property for partial differential equations.     

The derivative-dependent functional variable separation for the evolution equations

This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation u_tt=Au,u_xu_xx+Bu,u_x,u_t which admits the derivative-dependent functional separable solutions DDFSSs). We also extend the concept of the DDFSS to cover other variable separation approaches.     

A New Integrable Hierarchy,Parametric Solutions and Traveling Wave Solutions

In this paper we give a new integrable hierarchy. In the hierarchy there are the following representatives:

Analyzing powers Ayy , Axx , Axz and Ay in the dd → 3Hen reaction at 270 MeV

The data on the tensor A_yy, A_xx, A_xz and vector A_y analyzing powers in the dd → ^3Hen obtained at T _d = 270 MeV in the angular range 0^° - 110^° in the c.m. are presented. The observed negative sign of the tensor analyzing powers A_yy, A_xx and A_xz at small angles clearly demonstrate the sensitivity to the ratio of the D - and S -wave component of the ³He wave function. However, the one-nucleon exchange calculations by using the standard ³He wave functions have failed to reproduce the strong variation of the tensor analyzing powers as a function of the angle in the c.m.     

2-manifolds of constant curvature, 3-parameter isometry groups and bäcklund transformations

The relationship between the Sine Gordon equation and Riemannian 2-manifolds of constant negative curvature is well known. We point out that it gives rise (locally) to a transitive action of the 3-parameter group SL(2,R) on the manifold. This enables us to elucidate group theoretically the geometry underlying the Bäcklund transformation. The transformation arises by passing from the 2-manifold, a homogeneous space, to 2-submanifolds of SL(2,R) defined by orbits of the solvable subgroup of SL(2,R). A similar approach allows us several other such transformations, for example, transformations linking φ_xx+φ_yy = sinhφ to φ_xx+φ_yy = sinφ and φ_xx+φ_yy = e^2φ to φ_xx+φ_yy = 0.