ON THE PROLONGATION STRUCTURE OF ERNST EQUATION

An SL(2R) ×R¹(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is pfesented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neugebauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbertproblem (HHP), would be covered by this condition.     

Quantum Minkowski Space and Generators of Quantum Lorentz Group

From the basic 4 × 4 R matrix associated with the quantum Lorentz group SL_q(2, C) and its various fusion matrices, the covariant differential calculus on the quantum Minkowski space and the R commutation relation for the covariant generators of quantum Lorents group are presented.     

R'2 measured in trabecular bone in vitro: relationship to trabecular separation.

Measurement of key parameters of the microstructure of trabecular bone is critical to the study of osteoporosis and bone strength. Density based methods cannot provide this information, and give only the total amount of bone present, and not its arrangement. Magnetic resonance imaging has shown the potential to provide information related to the microarchitecture of the trabecular bone matrix. Twelve samples (8 x 8 x 8 mm3 bone cubes) were cut from sheep vertebrae such that the trabeculae ran either parallel or perpendicular to each face. Detailed measurements of the structure of these bone cubes were made by histomorphometry, and compared to R'2 and R*2 measured with a spin and gradient-echo sequence, Partially Refocused Interleaved Multiple Echo, at 1.5 Tesla. The precision of the R'2 measurement (% coefficient of variation) was 8.7+/-5.1, and 7.7+/-4.3 for R*2. Uncorrected values of R'2 and R*2 were significantly correlated to density measured by quantitative computed tomography (r = 0.87, p = 0.0005, and r = 0.90, p = 0.0002, respectively), and trabecular bone area measured by histomorphometry (r = 0.80, p = 0.002, and r = 0.83, p = 0.0008, respectively). Density correction was effected by imaging the same slice of bone in two orientations (90 degrees and 0 degrees ) to the main magnetic field. For both R'2 and R*2 there was a significant difference between measurements in the 90 degrees and 0 degrees orientations (p < 0.01). The difference between the two values was used, and termed R'2net or R*2net. The net parameters were independent of bone mass. R'2net and R*2net were significantly correlated to trabecular separation (p < 0.05) with r = -0.58 and r = -0.62, respectively. These results demonstrate the ability of magnetic resonance imaging to characterize a key measure of the trabecular microstucture. An increase in trabecular separation has important biomechanical consequences in osteoporosis. This result also strengthens the hypothesis that the sensitivity of R'2 to osteoporosis-related bone changes is due to magnetic susceptibility effects in which rapid transitions between bone and marrow create local magnetic field inhomogeneities that result in an increase in R'2 values.     

Relational Evolution of the Degrees of Freedom of Generally Covariant Quantum Theories

We study the classical and quantum dynamics of generally covariant theories with vanishing Hamiltonian and with a finite number of degrees of freedom. In particular, the geometric meaning of the full solution of the relational evolution of the degrees of freedom is displayed, which means the determination of the total number of evolving constants of motion required. Also a method to find evolving constants is proposed. The generalized Heisenberg picture needs M time variables, as opposed to the Heisenberg picture of standard quantum mechanics where one time variable t is enough. As an application, we study the parametrized harmonic oscillator and the SL(2, R) model with one physical degree of freedom that mimics the constraint structure of general relativity where a Schrödinger equation emerges in its quantum dynamics.     

Geometry of canonical variables in gravity theories

We show how to pass from an SL(2, C) covariant to an SU(2) covariant formulation of the theories of gravity. Our construction determines the canonical and gauge variables of the theory and establishes an appropriate framework for a hamiltonian picture.     

SU(2) Charges as Angular Momentum in N = 1 Self-Dual Supergravity

The N = 1 self-dual supergravity has SL(2, ) symmetry. This symmetry results inSU(2) charges as the angular momentum. As innonsupersymmetric self-dual gravity, the currents arealso of their potentials and are therefore identically conserved. Thecharges are generally invariant and gauge covariantunder local SU(2) transforms and approach being rigid atspatial infinity. The Poisson brackets constitute the su(2) algebra and hence can be interpretedas the generally covariant conservative angularmomentum.     

Noncommutative AKNS Equation Hierarchy and Its Integrable Couplings with Kronecker Product

We present a noncommutative version of the Ablowitz-Kaup-Newell-Segur （AKNS） equation hierarchy, which possesses the zero curvature representation. Furthermore, we derive the noncommutative AKNS equation from the noncommutative （anti-）self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, the integrable coupling system of the noncommutative AKNS equation hierarchy is constructed by using the Kronecker product.     

Integrable Deformations of the (2+1)-Dimensional Heisenberg Ferromagnetic Model

Based on the covariant prolongation structure technique,we construct the integrable higher-order deformations of the (2+1)-dimensional Heisenberg ferromagnet model and obtain their su(2)×R(λ) prolongation structures.By associating these deformed multidimensional Heisenberg ferromagnet models with the moving space curve in Euclidean space and using the Hasimoto function,we derive their geometrical equivalent counterparts,i.e.,higher-order (2+1)-dimensional nonlinear Schrödinger equations.     

On a generalized Ablowitz–Kaup–Newell–Segur hierarchy in inhomogeneities of media: soliton solutions and wave propagation influenced from coefficient functions and scattering data

In this paper, a generalized Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy in inhomogeneities of media described by variable coefficients is investigated, which includes some important nonlinear evolution equations as special cases, for example, the celebrated Korteweg–de Vries equation modeling waves on shallow water surfaces. To be specific, the known AKNS spectral problem and its time evolution equation are first generalized by embedding a finite number of differentiable and time-dependent functions. Starting from the generalized AKNS spectral problem and its generalized time evolution equation, a generalized AKNS hierarchy with variable coefficients is then derived. Furthermore, based on a systematic analysis on the time dependence of related scattering data of the generalized AKNS spectral problem, exact solutions of the generalized AKNS hierarchy are formulated through the inverse scattering transform method. In the case of reflectionless potentials, the obtained exact solutions are reduced to n-soliton solutions. It is graphically shown that the dynamical evolutions of such soliton solutions are influenced by not only the time-dependent coefficients but also the related scattering data in the process of propagations.     

Covariant Prolongation Structure of Konno-Asai-Kakuhata Equation

Based upon the covariant prolongation structures theory, we construct the sl(2,R)×R(ρ) prolongation structure for Konno-Asai-Kakuhata equation. By taking two and one-dimensional prolongation spaces, we obtain the inverse scattering equations given by Konno et al. and the corresponding Riccati equation. The Bäcklund transformations are also presented.     

A discrete negative AKNS equation: generalized Cauchy matrix approach

Generalized Cauchy matrix approach is used to investigate a discrete negative Ablowitz–Kaup–Newell–Segur (AKNS) equation. Several kinds of solutions more than multi-soliton solutions to this equation are derived by solving determining equation set. Furthermore, applying an appropriate continuum limit we obtain a semidiscrete negative AKNS equation and after a second continuum limit we derive the nonlinear negative AKNS equation. The reductions to discrete, semi-discrete and continuous sine-Gordon equations are also discussed.     

Dynamical analysis of diversity lump solutions to the (2+1)-dimensional dissipative Ablowitz–Kaup–Newell–Segure equation

The lump solution is one of the exact solutions of the nonlinear evolution equation. In this paper, we study the lump solution and lump-type solutions of (2+1)-dimensional dissipative Ablowitz–Kaup–Newell–Segure (AKNS) equation by the Hirota bilinear method and test function method. With the help of Maple, we draw three-dimensional plots of the lump solution and lump-type solutions, and by observing the plots, we analyze the dynamic behavior of the (2+1)-dimensional dissipative AKNS equation. We find that the interaction solutions come in a variety of interesting forms.     

Complete soliton solutions of the ZS/AKNS equations of the inverse scattering method

Starting with the (n-1)-soliton solution of a non-linear evolution equation (NLEE) and the corresponding Zakharov-Shabat, Ablowitz-Kaup-Newell-Segur (ZS/AKNS) wavefunctions, we obtain the n-soliton solution of the NLEE and the corresponding ZS/AKNS wavefunctions. This is then used to obtain complete soliton solutions of the NLEE and the ZS/AKNS equations.     

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding(2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation(BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing(2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the(2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the(2+1)-dimensional AKNS equation(also called the Davey-Stewartson hierarchy), a kind of(2+1)-dimensional Schr¨odinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new(2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the(2+1)-dimensional integrable coupling, which is further reduced to the standard(2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known(1+1)-dimensional AKNS hierarchy, the(1+1)-dimensional nonlinear Schr¨odinger equation are all special cases of the(2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the(2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated.     

Absence of Subsystems for the Haag–Kastler Net Generated by the Energy-Momentum Tensor in Two-Dimensional Conformal Field Theory

We show that if A is the Haag–Kastler net generated by the energy-momentum tensor in a chiral quantum field theory, then every subsystem B A which is covariant under the action of SL(2,R given on A must coincide with A. The result is valid for all the allowed values of the central charge and is obtained using scaling limit techniques.     

Local currents for the GL(N,C) self-dual Yang-Mills equation

By using a simple Bäcklund-like transformation which linearizes the GL(N, C) self-dual Yang-Mills equation, an infinite number of local conservation laws for this equation are constructed. In the SL(N, C) case, the currents become trivial, which explains why these currents are not found in SU(N) gauge theory.     

Critical Behavior of the Generalized Srnoluchovski's Coagulation Equation

The critical behavior of the gelation of polymers is studied by means of the generalized Srnoluchovski's coagulation equation. The exact solution of the kinetic equation with a factorial coagulation rate R(i₁, i₂, . . . i_n) = s_i1 × s_i2 × . . . × s_in and s_k = A × k + B is derived for the monodisperse initial condition c_m(O) = δ_{m, 1}. It is shown that a gelation transition takes place within a finite time t_c and the gelation time t_c ii characterized by the parameters A and B.     

Joint transfer of time and frequency signals and multi-point synchronization via fiber network

A system of jointly transferring time signals with a rate of 1 pulse per second(PPS) and frequency signals of 10 MHz via a dense wavelength division multiplex-based(DWDM) fiber is demonstrated in this paper.The noises of the fiber links are suppressed and compensated for by a controlled fiber delay line.A method of calibrating and characterizing time is described.The 1PPS is synchronized by feed-forward calibrating the fiber delays precisely.The system is experimentally examined via a 110 km spooled fiber in laboratory.The frequency stabilities of the user end with compensation are1.8×10~(-14) at 1 s and 2.0×10~(-17) at 10~4 s average time.The calculated uncertainty of time synchronization is 13.1 ps,whereas the direct measurement of the uncertainty is 12 ps.Next,the frequency and 1PPS are transferred via a metropolitan area optical fiber network from one central site to two remote sites with distances of 14 km and 110 km.The frequency stabilities of 14 km link reach 3.0×10~(-14) averaged in 1 s and 1.4×10~(-17) in 10~4 s respectively;and the stabilities of 110 km link are 8.3×10~(-14) and 1.7×10~(-17),respectively.The accuracies of synchronization are estimated to be 12.3 ps for the14 km link and 13.1 ps for the 110 km link,respectively.     

Darboux Transformation for a Negative Order AKNS Equation

Using a quasideterminant Darboux matrix, we compute soliton solutions of a negative order AKNS (AKNS($-$1)) equation. Darboux transformation (DT) is defined on the solutions to the Lax pair and the AKNS($-$1) equation. By iterated DT to K-times, we obtain multisoliton solutions. It has been shown that multisoliton solutions can be expressed in terms of quasideterminants and shown to be related with the dressed solutions as obtained by dressing method.     

INTEGRABILITY OF CERTAIN DEFORMED NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

A systematic investigation of certain higher order or deformed soliton equations with (1 + 1) dimensions, from the point of complete integrability, is presented. Following the procedure of Ablowitz, Kaup, Newell and Segur (AKNS) we find that the deformed version of Nonlinear Schrodinger equation, Hirota equation and AKNS equation admit Lax pairs. We report that each of the identified deformed equations possesses the Painlevé property for partial differential equations and admits trilinear representation obtained by truncating the associated Painlevé expansions. Hence the above mentioned deformed equations are completely integrable.