Bi-solitons for a class of non-linear equations with quadratic non-linearity. I

We consider the class of non-integrable, non-linear equations,

$\text{L}{}_{\mathrm{q}}\text{K=K}{}^2\text{, L}{}_{\mathrm{q}}\text{=? +}\text{∑}\text{1?i+j?q}\text{aij?}{}^{\mathrm{i}}{}_{\mathrm{x}}{}^{\mathrm{i}}\text{?}{}^{\mathrm{j}}{}_{\mathrm{t}}{}^{\mathrm{j}}\text{, ?≠0,}$

in 1+1 dimensions. We seek rational solutions K(ω₁,ω₂), which we call bi-solitons, with exponential type variables ω_i = exp(γ_ix + ρ_it). In this paper, we restrict to q = 2 and 3, and investigate the general q case in the following paper. We find that these bi-solitons exist when the operator L_q (with ± ?) can be factorized as the product of smaller order differential operators. Besides the trivial factorized bi-solitons, we show that there exist non-trivial ones whenever K may be written as Σ^lmaxx ω^l₂F_l(Z = ω₁ + ω₂). In order to understand the origin of the factorization property, to any polynomial K = Σω^l₂F_l(Z) we associate a linear transformation such that L_qK has only the power ω^l₂ of K². For q = 2 and 3, we find that there exist particular polynomials of this type restraining L_q to be a product of smallr order operators. For the full non-linear equations we verify that all the bi-solitons can be obtained from these particular polynomials.     

Bi-solitons for a class of non-integrable non-linear partial differential equations with quadratic non-linearity

Investigating the bi-solitons of the class of equations (e + a?_x + b?_t + c?²_xt + d?²_xx)K = K², e ≠ 0, we report that they exist if and only if the operator of the linear part for K or e + K can be factorized.     

Pulsed EPR spectroscopy of Gd3+ centers in Ca1−x R x F2+x (R=La, Y) mixed crystals

A study of Gd³⁺ centers in Ca_1?x R _x F_2+x (R=La, Y) crystals using pulsed EPR spectroscopy is presented. The echo-induced EPR (ESE-EPR) spectrum shows, besides the signal of slightly perturbed cubic Gd³⁺ centers, a broad signal at g≈2 due to Gd³⁺ centers at low symmetry sites. To describe the effects of R³⁺ ions on the EPR Gd³⁺, a model, including cubic and linear R³⁺?2F _i ^? centers, is developed. Its predictions are compared with the experimental results. The composition dependence of the EPR signal due to slightly perturbed cubic Gd³⁺ centers in mixed Ca_1?x R _x F_2+x crystals is explained taking into account the different clustering tendency in La and Y crystals. Moreover, the formation of mixed clusters involving R³⁺ and Gd³⁺ ions is proposed for both series of samples. A greater clustering trend is found in the Y crystals than in the La ones. Gd³⁺ ions are found to be a “non innocent” paramagnetic probe for structural studies in these mixed crystals.     

Non-linear fluorescence in dye solutions induced by a low power laser

Measurements of the fluorescence intensity I_F, excited by two-photon pumping, have been carried out in different dye solutions by utilizing a low power cw laser (I_L ≈ 12 mW) and a particular experimental technique. A considerable departure from the quadratic law I_F ∝ I²_L with the varying of 2hω - ΔE_i quantity has been detected. This behaviour has bee n accounted for by considering the dependence of the non-linear and linear cross sections relative to the S₀ → S₁ and S₀ → S₂ transitions on the laser frequency.     

Rotating frames in special relativity

A uniformly rotating frame is defined as the rest frame of a particle revolving with constant velocityω in a circle about theZ-axis of an inertial frame Σ⁰. Under the conditionz=Z,r=R, theoretical constraints are established for the solution of the transformation problem Σ⁰→Σ^ω r,Σ^ω r being the cylindrical subframe of Σ^ω. The unique solution of the problem in cylindrical coordinates is isomorphic to the special Lorentz transformationL _x, withβ=v/c replaced byβ _r=ωr/c. Hence the intrinsic geometry on the surface of a rotating cylinder is Euclidean. Though there exists no complete intrinsic geometry on the surface of a rotating disk, the geodesics on it are straight lines while the circumference of a concentric circle isK _r2πr as predicted by Einstein.     

Degree Complexity of Birational Maps Related to Matrix Inversion

For a q × q matrix x = (x _{i, j} ) we let ${J(x)=(x_{i,j}^{-1})}For a q × q matrix x = (x _{i, j} ) we let J(x)=(x_i,j^-1){J(x)=(x_{i,j}^{-1})} be the Hadamard inverse, which takes the reciprocal of the elements of x. We let I(x)=(x_i,j)^-1{I(x)=(x_{i,j})^{-1}} denote the matrix inverse, and we define K=I°J{K=I\circ J} to be the birational map obtained from the composition of these two involutions. We consider the iterates Kⁿ=K°?°K{K^n=K\circ\cdots\circ K} and determine the degree complexity of K, which is the exponential rate of degree growth d(K)=lim_n?￥( deg(Kⁿ) )^1/n{\delta(K)=\lim_{n\to\infty}\left( deg(K^n) \right)^{1/n}} of the degrees of the iterates. Earlier studies of this map were restricted to cyclic matrices, in which case K may be represented by a simpler map. Here we show that for general matrices the value of δ(K) is equal to the value conjectured by Anglès d’Auriac, Maillard and Viallet.     

Investigations of EPR g Factors and Rhombic Distortion for the Rhombic Cu2+ Centers in K2Zn1?x Cu x F4 Crystals at Low Temperature

The electron paramagnetic resonance g factors g _i (i?=?x, y, z) of two rhombic Cu²⁺ centers, Cu²⁺(II) and Cu²⁺(III), in K₂Zn_1?x Cu _x F₄ crystals found at low temperature are calculated from the complete diagonalization (of energy matrix) method based on the cluster approach. The calculations show that the ground state wave function of the two rhombically compressed Cu²⁺ centers is $ | {d_{{z^{2} }} }\!\!> $ with a small admixture of $ | {d_{{x^{2} - y^{2} }} } \!\!> $ . The rhombic distortions for both Cu²⁺ centers are obtained from the calculations. The results are discussed.     

On asymptotic behaviour of four spin correlation functions in a linear chain of spins

Second and fourth moments of the four spin frequency correlation functions S^Z_i(t) S^Z_i+1 (t) S^Z_j+1 〉_k,ω and 〈S^x_i(t) S^z_i+1 (t) S^x_jS^z_j+1 〉_kω have been calculated in the high temperature limit for an arbitrary spin with an isotropic Heisenberg Hamiltonian.     

Simple tests of quark-nucleon reciprocity

The inclusive cross-section for the production of a single hadron in deep inelastic electroproduction is studied in a dual resonance model. The Bjorken scaling behaviour in the virtual photon fragmentation region for finite x (≡ 2p_L^c.m./√s) is (1/σ^{T,L))d3σT,L/E?}d³p ～ (1/q²) F(x,p_⊥²/q²) and thus the transverse momentum grows like q², whereas in the parton model (1/σ^T,L)d³σ^T,L/E^?1d³p ～ F(x,p_⊥²). A related effect is the absence of two-jet structure in e⁺e^? annihilation. We believe that dual model results may give a more reliable indication of the deep inelastic behaviour for composite hadrons than the parton model.     

Bounds for Kℓ3 decay parameters

The extremum of the integral ∫^∞f(x) ‖F (x)‖² dx is determined for the case when the values taken by the function F (x)¹ at three points are known. The result is applied to the K_?3 problem, the function F (x) being replaced by the form factor f₊(q²)+f₋(q²)q²(m_K²−m_π²) of the divergence of the strangeness changing vector current.     

Coagulation equations with gelation

Smoluchowski's equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernelK _ij models the bonding mechanism. For different classes of kernels we derive criteria for the occurrence of gelation, and obtain critical exponents in the pre- and postgelation stage in terms of the model parameters; we calculate bounds on the time of gelationt _c , and give an exact postgelation solution for the modelK _ij =(ij ^ω ) (ω>1/2) andK _ij =a ^i+j (a>1). For the modelK _ij =i ^ω +j ^ω (ω<1, without gelation) initial solutions are given. It is argued that the kernelK _ij ～ij ^ω with ω?1?1/d (d is dimensionality) effectively models the sol-gel transformation in polymerizing systems and approximately accounts for the effects of cross-linking and steric hindrance neglected in the classical theory of Flory and Stockmayer (Ω=1). For allΩ the exponents,t=Ω+3/2 andσ=Ω?1/2,γ=(3/2?Ω)/(Ω ? 1/2) andΒ=1, characterize the size distribution, at and slightly below the gel point, under the assumption that scaling is valid.     

Morphogenesis of sliced N-soliton solutions

Consider an instantaneous severing interaction which at t = t_s transforms is given N-soliton solution q⁰ into two new solutions, q^L and q^R, with discontinuous anitial conditions at t = t_s such that q^L(q^R) is equal to q⁰ to the left (right) of the severing point x_s and vanishes to the right (left) of x_s. The soliton content of q^L and q^R is studied in the context of the cubic Schrödinger equation and the sine-Gordon equation by solving the direct scattering problem. It is shown that the motion constants C_n, n = 1, 2 …, are related by , Furthermore, if q⁰ contains N solitons and no contribution from the continuous spectrum, then q^L(q^R) contains N^L(N^R) solitons with 0 ? N^L, N^R ? N and N^L + N^R = N. The slicing puts a soliton into q^L or q^R, respectively as the slicing point x_s is taken to the right or left of the soliton's “center” at t = t_s.     

The local structure of the spectrum of the one-dimensional Schrödinger operator

Let \(H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )\) be an one-dimensional random Schrödinger operator in ?²(?V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x _t ), wherex _t is a Brownian motion on the compact Riemannian manifoldK andF:K→R ¹ is a smooth Morse function, \(\mathop {\min }\limits_K F = 0\) . Let \(N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 \) , where Δ∈(0, ∞),E _i (V) are the eigenvalues ofH _V . The main result (Theorem 1) of this paper is the following. IfV→∞,E ₀>0,k∈Z ₊ anda>0 (a is a fixed constant) then $$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$ wheren(E ₀) is a limit state density ofH _V ,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH _V ,V→∞. The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.     

Inclusive φ-meson production in 93 and 63 GeV hadron interactions

The inclusive reactions h+p→φ+X, (h=π^±,,K^±,p^±), are studied for 0?x_F?0.3 and p⊥ ? 1 GeV at 93 and and 63 GeV incident momentum. Differential cross sections dσ/dp_⊥² anddσ/dx_F are presented and are compared with predictions of the naive parton model.     

Analyticity properties and a convergent expansion for the inverse correlation length of the high-temperatured-dimensional Ising model

We show that the inverse correlation lengthm(β) (= mass of the fundamental particle of the associated lattice quantum field theory) of the spin-spin correlation function 〈s _x s _y 〉,x, y εZ ^d , of thed-dimensional Ising model admits the representation $$m(\beta ) = - ln\beta + r(\beta )$$ for small inverse temperaturesβ > 0.r(β) is ad-dependent function, analytic atβ = 0.c _n , the nth β = 0 Taylor series coefficient of r(β) can be computed explicitly from the Z^d limit of a finite number of finite lattice A spin-spin correlation functions 〈s₀s_x〉t>Afor a finite number ofx = (x ₁,x₂, ..., x_d), ¦x¦ = ∑ _i ^d 1¦x_i¦< R(n), where R(n) increases withn. Furthermore, there exists aβ' > 0, such that for eachβ ε (0,β')m(β) is analytic. Similar results are also obtained for the dispersion curve ω(p), ω(p)=ω(0)=m, pε(-π, π]^d?1, of the fundamental particle of the associated lattice quantum field theory.     

Evolution des proprietes de transport des solutions solides M1−xM′xF2+x (M=Sr,Pb; M′=Sb,Bi) ET M1−xM″xF2+2x (M″=Zr,Th) pour les faibles taux de substitution

A detailed study of the transport properties of the solid solutions M_1?xM′_xF_2+x (M=Sr, Pb; M′=Sb, Bi) and M_1?xM″_xF_2+2x (M″=Zr, Th) has been carried out for x<0.10. Contrary to solid solutions of strontium, solid solutions of lead show for a very small value of x (x = 0.10 forPb_1?xM′_xF_2+x, x = 0.005 for Pb_1?xM″'_xF_2+2x) a minimum of conductivity associated with a maximum of activation energy. Hypotheses are proposed to explain the results.     

标定粉末照相指数的一个新图解法

本文叙述了标定德拜-谢乐照相指数的一个新图解法。利用三条低角度衍射线的sin²θ值和该晶体的密度,在A-C空间画一系列条件直线及等原子曲线。三条条件直线和一条等原子曲线的交点直接决定了晶胞的大小和晶胞内所含的原子数或位形单位数。这个方法可应用于四方晶系和六角晶系。 关键词：     

Fast ion conduction in the defect pyrochlore system KBxW2−xO6(B = Al,Ti and Ta)

K⁺ ionic conductivity has been measured by the complex impedence method for the defect pyrochlore type compounds of the general formula K(B^y+_xW⁶⁺_2?x)O₆, (x=1/(6?y)) where B=Al³⁺, Ti⁴⁺ and Ta⁵⁺. Thermogravimetric analysis showed the compounds to be hydrated at room temperature with from ∽0.2 mole for the Al compound to ∽1 mole of water for the Ta substituted phase. There is a correlation between the lattice parameter and amount of water present in the cavities. The ionic conductivity of 5.2×10^?3ω^?1cm^-1 at 300°C for KAl_0.33W_1.67O₆ is found to be best among known KB₂O₆ type pyrochlores. High ionic conductivity has been explained on the basis of occupancy of K⁺ ions in the available sites and polarization of the B₂X₆ network.     

Synthesis,crystal structure,and luminescence properties of CaY2Ge3O10:Ln3+, Ln = Eu,Tb

The crystal structure and luminescence properties of CaY₂Ge₃O₁₀:Ln³⁺ (Ln = Eu, Tb) germanates synthesized via a conventional solid-state reaction and an ethylenediaminetetraacetic acid complexing process are studied. The CaY_{2 ? x} Ln _x Ge₃O₁₀ (Ln = Eu, Tb; x = 0–1.0, 2.0; Δx = 0.1) solid solutions have a monoclinic structure (space group P2₁/c, Z = 4), in which dopant ions occupy three nonequivalent noncentrosymmetric sites with different Ca²⁺/Ln³⁺ ratios. The effect of the synthesis methods, dopant concentrations, and excitation wavelengths on the luminescence properties of the compounds obtained is determined.     

Anionic conductivity and thermal stability of single crystals of solid solutions based on calcium fluoride

The temperature and composition dependencies of the anionic conductivity in the temperature range from ambient to 1073 K were studied for single crystals of Ca_1−xGd_xF_2+x (x=1×10⁻⁴, 1×10⁻³, 1×10⁻² and 1×10⁻¹) and of Ca_0.8R_0.2F_2.2 (R=La, Ce, Pr, Nd, Gd, Dy, Er, Tm, Yb, Lu and Y), having the fluorite structure. The conductivity plots for the concentrated Ca_0.8R_0.2F_2.2 solid solutions display the low-temperature and high-temperature linear (Arrhenius) regions with the knee temperature T_k∼770 K. The values of the conductivity activation enthalpy obey the relation ΔH_HT(T>T_k)>ΔH_LT(T<T_k). The conductivity mechanism in heavily doped Ca_1−xR_xF_2+x crystals is associated with the clusters of the point defects which decrease the potential barriers for fluoride anions moving by hops over the structural sites of the anion sub-lattice. We studied the effect of the dimensional factor (doped cation radii) on the anionic transport in these crystals. A correlation between anionic transport and atomic structure of the studied crystals is discussed.