A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties of the meromorphic functions, including asymptotic expansions at infinite points, we obtain the essential singularities and divisor of the Baker-Akhiezer function. Utilizing the theory of algebraic curves, quasi-periodic solutions for the entire hierarchy are finally derived in terms of the Riemann theta function.     

A note on the quasi-periodic solutions of the modified Boussinesq hierarchy

Based on the theory of trigonal curve and the properties of three kinds of the Abel differentials on it, we deduce the explicit theta function representations of the Baker-Akhiezer function and the meromorphic function associated with the modified Boussinesq hierarchy. The modified Boussinesq flows are straightened using the Abel map and the Lagrange interpolation formula. The explicit theta function representations of solutions for the entire modified Boussinesq hierarchy are constructed with the aid of the asymptotic properties and the algebro-geometric characters of the meromorphic function.     

Algebro-geometric Constructions of Quasi Periodic Flows of the Discrete Self-dual Network Hierarchy and Applicationsℰ

In this paper we obtain the discrete integrable self-dual network hierarchy associated with a discrete spectral problem. On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the discrete self-dual network hierarchy are straightened using the Abel-Jacobi coordinates. The meromorphic function and the Baker-Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the discrete self-dual network hierarchy are constructed with the help of the asymptotic properties and the algebra-geometric characters of the meromorphic function, the Baker-Akhiezer function and the hyperelliptic curve.     

Algebro-Geometric Solutions of the TD Hierarchy

Resorting to the Lenard recursion scheme, we derive the TD hierarchy associated with a 2?×?2 matrix spectral problem and establish Dubrovin-type equation in terms of the introduced elliptic variables. Based on the theory of algebraic curve, all the flows associated with the TD hierarchy are straightened under the Abel-Jacobi coordinates. An algebraic function ?, also called the meromorphic function, carrying the data of the divisor is introduced on the underlying hyperelliptic curve $\mathcal {K}_{n}$ . The known zeros and poles of ? allow to find theta function representations for ? by referring to Riemann’s vanishing theorem, from which we obtain algebro-geometric solutions for the entire TD hierarchy with the help of asymptotic expansion of ? and its theta function representation.     

Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire HS2 hierarchy.     

Vibrational modes of the stibine molecule

In this paper, we use the algebraic approach to describe the vibrational modes of stibine molecule (of C_3v molecular symmetry group) up to 21 quanta. As the stibine molecule exhibits stretch-bend resonances, we build an algebraic pyramidal coupling operator between stretching modes and bending modes adapted to this molecule. The standard deviation associated to the fit of the vibrational levels is 1.75 cm⁻¹.     

Nahm Transform and Spectral Curves¶for Doubly-Periodic Instantons

We present the Nahm transform of the doubly-periodic instantons (i.e. instantons on T ²×ℝ²), converting them into certain meromorphic solutions of Hitchin's equations over an elliptic curve. We then show how to construct a triple consisting of an algebraic curve plus a line bundle with connection over it from a doubly-periodic instanton, and that such data coincides with the Hitchin spectral data associated with the Nahm transformed Higgs bundle. Received: 15 October 1999 / Accepted: 16 October 2001     

Line mixing effects in the ν2 + ν3 band of methane

This study provides the first direct experimental measurements of the off-diagonal relaxation matrix element coefficients for line mixing in air-broadened methane spectra for any vibrational band and the first off diagonal relaxation matrix elements associated with line mixing for pure methane in the ν₂ + ν₃ band of ¹²CH₄. The speed-dependent Voigt profile with line mixing is used with a multispectrum nonlinear least squares curve fitting technique to retrieve the various line parameters from 11 self-broadened and 10 air-broadened spectra simultaneously. The room temperature spectra analyzed in this work are recorded at 0.011 cm⁻¹ resolution with the McMath-Pierce Fourier transform spectrometer located at the National Solar Observatory, Kitt Peak, Arizona. The off-diagonal relaxation matrix element coefficients of ν₂ + ν₃ transitions between 4410 and 4629 cm⁻¹ are reported for eighteen pairs with upper state J values between 2 and 11. The observed line mixing coefficients for self broadening vary from 0.0019 to 0.0390 cm⁻¹ atm⁻¹ at 296 K. The measured line mixing coefficients for air broadening vary from 0.0005 to 0.0205 cm⁻¹ atm⁻¹ at 296 K.     

On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations

Recently, considerable progress has been made in understanding the nature of the algebro-geometrical superposition principles for the solutions of nonlinear completely integrable evolution equations, and mainly for the equations related to hyperelliptic Riemann surfaces. Here we find such a superposition formula for particular real solutions of the KP and Boussinesq equations related to the nonhyperelliptic curve ⁴ = ( – E ₁) ( – E ₂) ( – E ₃) ( – E ₄). It is shown that the associated Riemann theta function may be decomposed into a sum containing two terms, each term being the product of three one-dimensional theta functions. The space and time variables of the KP and Boussinesq equations enter into the arguments of these one-dimensional theta functions in a linear way.On leave from Leningrad State University and Leningrad Institute of Aviation Instrumentation.     

A nonperturbative solution to the Dyson-Schwinger-equations of QCD

This is the first of two papers in which we discuss a nonperturbatively modified solution to the Euclidean Dyson-Schwinger equations for the 7 superficially divergent proper verticesΓ of QCD. It takes the formΣ _n g ²ⁿ Γ( ⁿ ) where eachΓ( ⁿ ) approaches its perturbative form at large momenta. At lower momenta, it differs from that form by an additional non-analyticg ² dependence through a dynamical mass scaleb, proportional toΛ _qcd and associated with a pole dependence on the momentum invariants. In the zeroth-order two-point functions, these nonperturbative modifications amount to a generalized Schwinger mechanism, leading to propagators without particle poles. The termsΓ(⁰), representing the Feynman rules of the modified iterative solution, can become self-consistent in the DS equations through a mechanism of “nonperturbative logarithms” which we explain. The mechanism is tied to the presence of divergent loops, and thus represents a pure quantum effect, similar to quantum anomalies. It restricts formation of nonperturbativeΓ(⁰)'s to the 7 primitively divergent vertices, thus escaping the infinite nature of the DS hierarchy. In a given loop order, the self-consistency problem reduces to a finite set of algebraic equations.     

New binary travelling-wave periodic solutions for the modified KdV equation

In this Letter, the modified Korteweg-de Vries (mKdV) equations with the focusing (+) and defocusing (−) branches are investigated, respectively. Many new types of binary travelling-wave periodic solutions are obtained for the mKdV equation in terms of Jacobi elliptic functions such as sn(ξ,m)cn(ξ,m)dn(ξ,m) and their extensions. Moreover, we analyze asymptotic properties of some solutions. In addition, with the aid of the Miura transformation, we also give the corresponding binary travelling-wave periodic solutions of KdV equation.     

Quasiperiodic Solutions of the Heisenberg Ferromagnet Hierarchy

We present quasi-periodic solutions in terms of Riemann theta functions of the Heisenberg ferromagnet hierarchy by using algebro-geometric method. Our main tools include algebraic curve and Riemann surface, polynomial recursive formulation and a special meromorphic function.     

Compound windows of the Hénon-map

For the two-parameter second-order Hénon map, the shapes and locations of the periodic windows-continua of parameter values for which solutions x₀,x₁,… can be stably periodic, embedded in larger regions where chaotic solutions or solutions of other periods prevail-are found by a random searching procedure and displayed graphically. Many windows have a typical shape, consisting of a central “body” from which four narrow “antennae” extend. Such windows, to be called compound windows, are often arranged in bands, to be called window streets, that are made up largely of small detected but poorly resolved compound windows.For each fundamental subwindow-the portion of a window where a fundamental period prevails-a stability measure U is introduced; where the solution is stable, |U|<1. Curves of constant U are found by numerical integration. Along one line in parameter space the Hénon-map reduces to the one-parameter first-order logistic map, and two antennae from each compound window intersect this line. The curves where U=1 and U=−1 that bound either antenna are close together within these intersections, but, as either curve with U=−1 leaves the line, it diverges from the curve where U=1, crosses the other curve where U=−1, and nears the other curve where U=1, forming another antenna. The region bounded by the numerically determined curves coincides with the subwindow as found by random searching. A fourth-degree equation for an idealized curve of constant U is established.Points in parameter space producing periodic solutions where x₀=x_m=0, for given values of m, are found to lie on Cantor sets of curves that closely fit the window streets. Points producing solutions where x₀=x_m=0 and satisfying a third condition, approximating the condition that x_n be bounded as n→−∞, lie on curves, to be called street curves of order m, that approximate individual members of the Cantor set and individual window streets. Compound windows of period m+m^′ tend to occur near the intersections of street curves of orders m and m^′.Some exceptions to what appear to be fairly general results are noted. The exceptions render it difficult to establish general theorems.     

TiO2 activation using acid-treated vermiculite as a support: Characteristics and photoreactivity

Vermiculite was treated by sulfuric or nitric acid aqueous solutions with different concentration. These modified materials as the promising supports, were used to immobilize TiO₂. TiO₂ was prepared by the precursor, which was obtained by substituting partly isopropyl alcohol with Cl⁻ in titanium chloride {[Ti(IV)(OR)_nCl_m] (n = 2-3, m = 4 − n)}. The TiO₂/vermiculite composites were characterized by X-ray diffraction, scanning electron microscopy, and the nitrogen absorption. Their photocatalytic activity was evaluated by removal of methylene blue (MB). The pure anatase type crystalline phase was well deposited on the supports. The concentrations of acid for treatment had a significant influence on pore sizes and surface area of vermiculite. The treatment process changed microstructure of vermiculite, modified its characteristics, and farther improved the catalytic activity and absorption capacity of TiO₂/vermiculite composites. The treatment effect of nitric acid was superior to that of sulfuric acid.     

Temperature dependence of magnetic anisotropy constant in cobalt ferrite nanoparticles

The temperature dependence of the effective magnetic anisotropy constant K(T) of CoFe₂O₄ nanoparticles is obtained based on the SQUID magnetometry measurements and Mössbauer spectroscopy. The variation of the blocking temperature T_B as a function of particle radius r is first determined by associating the particle size distribution and the anisotropy energy barrier distribution deduced from the hysteresis curve and the magnetization decay curve, respectively. Finally, the magnetic anisotropy constant at each temperature is calculated from the relation between r and T_B. The resultant effective magnetic anisotropy constant K(T) decreases markedly with increasing temperature from 1.1×10⁷ J/m³ at 5 K to 0.6×10⁵ J/m³ at 280 K. The attempt time τ₀ is also determined to be 6.1×10⁻¹² s which together with the K(T) best explains the temperature dependence of superparamagnetic fraction in Mössbauer spectra.     

Magnetic properties of (Nd,Pr)–Fe–B nanocomposite over quenched and annealed ribbons

The base alloys of nominal composition (Nd_0.75Pr_0.25)_yFe_balanceB_x (y=10−9.2 and x=6−19.2) were chosen to study the influence of RE/B ratio, smaller than stochiometric composition on magnetic properties of over quenched and annealed ribbons. From X-ray diffraction analysis of these ribbons, the α-Fe and Fe₃B phases were observed along with (Nd,Pr)₂Fe₁₄B major phase. The average grain size was calculated using these patterns as: 35 nm for α-Fe, 45 nm for (Nd,Pr)₂Fe₁₄B and 22 nm for Fe₃B particles. TEM analysis also supported the nano distribution of the above phases. These X-ray graphs support the idea of exchange coupling between hard and soft phases responsible for the observed magnetic properties. In these ribbons the saturation magnetization J_s and remnant magnetization J_r increases from 1.19 T to 1.66 T and from 0.65 T to 0.91 T, respectively as RE/B ratio increases. The increase in J_s and J_r may be attributed to the presence of exchange coupling between these phases. The corresponding coercivity _jH_c decreases from 673.33 to 271.33 k Am⁻¹. The maximum energy product (BH)_max initially increases from 72.42 kJ m⁻³to 109.85 kJ m⁻³ up to RE/B≈1 and then decreases to 58.5 kJ m⁻³, depending on the shape of second quadrant B–H loop. The coercivity mechanism observed from initial hysteresis curve was considered to be nucleation of domain wall.     

Realizing tri-bimaximal mixing in minimal seesaw model with S4 family symmetry

In this Letter, we realize the tri-bimaximal mixing in the lepton sector in the context of minimal seesaw in which only two right-handed neutrinos are introduced, with the discrete group S4 as the family symmetry. In order to constrain the form of superpotential, a Z3 symmetry is also introduced. In the model, the mass matrices for charged leptons and right-handed neutrinos are diagonal. The unitary matrix that diagonalizes the light Majorana neutrino mass matrix is exact tri-bimaximal at LO, and is corrected by small quantities of O(0.01) at NLO. The mechanism to get the particular scalar VEV alignments used is also presented. Phenomenologically, the mass spectrum is of normal hierarchy with m₁=0, and ∑mi and |mee| are about 0.058 eV and 0.003 eV respectively.     

Predominant factors affecting the dielectric and piezoelectric properties of bismuth-containing complex perovskite solid solution

To study the factors affecting the dielectric and piezoelectric properties of bismuth-containing complex perovskites, the solid solution (1−x)Pb(Mg_1/3Nb_2/3)O₃-xBi(Mg_2/3Nb_1/3)O₃ was prepared by the solid state reaction method and its dielectric and piezoelectric properties were investigated. It is found that (1) at room temperature, the nonlinearity of the DE-loop for Pb(Mg_1/3Nb_2/3)O₃ is completely suppressed at a rather low x (<5%); (2) dielectric constant versus temperature curves deviate from the Curie-Weiss law at a temperature T_d much higher than the dielectric constant peak temperature T_m and T_m−T_d decreases considerably with increasing x; and (3) frequency dispersion ΔT_m=T_m (1 MHz)−T_m (10 kHz) increases with increasing x. Possible factors responsible for the variation of the dielectric and piezoelectric properties with x are discussed.     

Scaling analysis of phase transition in Hg-based superconductor

With the vibrating-reed technique, the internal friction (IF) Q⁻¹ is measured for sing-phase (Hg_0.66Pb_0.34)Ba₂Ca₂Cu₃O_8+x superconductor as a function of temperature at low applied magnetic field up to 0.5 T and as a function of frequency at normal state temperatures. An IF peak associated with flux motion can be found below T_C. The IF peak becomes higher and shifts towards lower temperature with increasing magnetic field. In addition an IF peak is found near 200 K. By scaling analysis we have demonstrated that the internal friction around the peak temperature can be collapsed into a single curve, indicating that the IF peak below T_C is originated from a phase transition associated with a vortex glass transition and a structural phase transition occurs at around 200 K in the superconductor.     

Formation mechanism of ferromagnetism in Si1−xMnx diluted magnetic semiconductors

Si_1−xMn_x diluted magnetic semiconductor (DMS) bulks were formed by using an implantation and annealing method. Energy dispersive X-ray fluorescence, transmission electron microscopy (TEM), and double-crystal rocking X-ray diffraction (DCRXD) measurements showed that the grown materials were Si_1−xMn_x crystalline bulks. Hall effect measurements showed that annealed Si_1−xMn_x bulks were p-type semiconductors. The magnetization curve as a function of the magnetic field clearly showed that the ferromagnetism in the annealed Si_1−xMn_x bulks originated from the interaction between interstitial and substitutional Mn⁺ ions, which was confirmed by the DCRXD measurements. The magnetization curve as a function of the temperature showed that the ferromagnetic transition temperature was approximately 75 K. The present results can help to improve understanding of the formation mechanism of ferromagnetism in Si_1−xMn_x DMS bulks.