A generalized O(2,1) expansion in the space-space region

Following the approach of Jones, Low and Young, a generalized O(2, 1) expansion is developed for amplitudes that have a power bounded growth asymptotically. The expansion, set up in an O(1, 1) basis, holds in a new kinematical region, where all the incoming and outgoing clusters have space-like O(2, 1) momenta.     

On theO(1/N 2)β-function of the Nambu-Jona-Lasinio model with non-abelian chiral symmetry

We present the formalism for computing the critical exponent corresponding to the β-function of the Nambu-Jona-Lasinio model withSU(M)×SU(M) continuous chiral symmetry atO(1/N ²) in a largeN expansion, whereN is the number of fermions. We find that the equations can only be solved for the caseM=2 and subsequently an analytic expression is then derived. This contrasting behavior between theM=2 andM>2 cases, which appears first atO(1/N ²), is related to the fact that the anomalous dimensions of the bosonic fields are only equivalent forM=2.     

Tensor representations of conformal algebra and conformally covariant operator product expansion

A conformally covariant formulation of operator product expansion is discussed as an expansion of the product of two representations into a direct sum of irreducible representations. The basic irreducible representations are analyzed and classified. The isomorphism between the conformal algebra and the O(4, 2) algebra is used to obtain a manifestly covariant formalism. The implications of the isomorphism in the derivation of the representations is discussed. The covariant O(4, 2) formalism directly relates dominant terms to nondominant terms in the light-cone limit. The essential coincidence of the problem of a conformal covariant operator product expansion to the problem of determining the form of the three-point function is stressed, together with the relevance of a selection rule for two-point functions following from exact (not spontaneously broken) conformal covariance. The role of Ward identities in a conformal covariant scheme is pointed out, and the mathematical implications on the n-point functions from causality are described.     

A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations

With the aid of symbolic computation, a generalized auxiliary equation method is proposed to construct more general exact solutions to two types of NLPDEs. First, we present new family of solutions to a nonlinear Klein-Gordon equation, by using this auxiliary equation method including a new first-order nonlinear ODE with six-degree nonlinear term proposed by Sirendaoreji. Then, we apply an indirect F-function method very close to the F-expansion method to solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l,n,p). Taking advantage of the new first-order nonlinear ODE with six degree nonlinear term, this indirect F-function method is used to map the solutions of C(l,n,p) equations to those of that nonlinear ODE. As a result, we can successfully obtain in a unified way, many exact solutions.     

Multiscale derivation of an augmented Smoluchowski equation

Smoluchowski and Fokker-Planck equations for the stochastic dynamics of order parameters have been derived previously. The question of the validity of the truncated perturbation series and the initial data for which these equations exist remains unexplored. To address these questions, we take a simple example, a nanoparticle in a host medium. A perturbation parameter ε, the ratio of the mass of a typical atom to that of the nanoparticle, is introduced and the Liouville equation is solved to O(ε²). Via a general kinematic equation for the reduced probability W of the location of the center-of-mass of the nanoparticle, the O(ε²) solution of the Liouville equation yields an equation for W to O(ε³). An augmented Smoluchowski equation for W is obtained from the O(ε²) analysis of the Liouville equation for a particular class of initial data. However, for a less restricted assumption, analysis of the Liouville equation to higher order is required to obtain closure.     

On the possibility of an analytic solution of the three-body problem

Three-body Faddeev equations in the Noyes-Fiedeldey form are rewritten as a matrix analog of a one-dimensional nonrelativistic Schrödinger equation. Unlike the method of K-harmonics, where a similar equation was obtained by expansion of a three-body Schrödinger equation wavefunction into the orthogonal set of functions of two variables (K-harmonics), the use of the Noyes-Fiedeldey form of Faddeev equations allows us to limit ourselves to the expansion in functions of one variable only. The solutions of the above mentioned matrix equation are obtained. These solutions converge uniformly within every interval of continuity of the matrix, which corresponds to the potential of that equation. Their asymptotic behavior for large interparticle distances is discussed. The solutions for the harmonic oscillator, inverse-square, and Coulomb-Kepler potentials are found. It is shown that energy levels in the last case may be calculated from a simple formula which is very similar to the corresponding formula for the two-body Coulomb-Kepler problem. This formula can be easily generalized to the case of n particles interacting with inverse distance potentials.     

Time-dependent realization of the infinite-dimensional hydrogen algebra

We use the Hamiltonian formalism to investigate the Katzin–Levine model of a time-dependent Kepler problem. This formalism enables us to define Lie products in terms of Poisson brackets and obtain a time-dependent realization of centerless twisted (or standard) Kac–Moody algebras of so(N+1). We also show that the classical solutions of the model are modulated conic sections and derive a generalized Kepler equation for the time dependence.     

The Fokker–Planck–Kolmogorov Equation with Nonlocal Nonlinearity in the Semiclassical Approximation

A scheme for constructing quasi-classical concentrated solutions of the Fokker–Planck–Kolmogorov equation with local nonlinearity is presented on the basis of the complex WKB-Maslov method. Formal, asymptotic in a series expansion parameter D, D 0 solutions of the Cauchy problem for this equation are constructed with a power accuracy O(D ^3/2). A set of the Hamilton–Ehrenfest equations (a set of equations for average and centered moments) derived in this work is of considerable importance in construction of these solutions. An approximate Green's function is constructed and a nonlinear principle of superposition is formulated in the class of semiclassical concentrated solutions of the Fokker–Planck–Kolmogorov equations.     

On long-wave sound scattering by a Rankine vortex: Non-resonant and resonant cases

The well-known two-dimensional problem of sound scattering by a Rankine vortex at small Mach number M is considered. Despite its long history, the solutions obtained by many authors still are not free from serious objections. The common approach to the problem consists in the transformation of governing equations to the d’Alembert equation with right-hand part. It was recently shown [I.V. Belyaev, V.F. Kopiev, On the problem formulation of sound scattering by cylindrical vortex, Acoustical Physics 54(5) (2008) 603-614] that due to the slow decay of the mean velocity field at infinity the convective equation with nonuniform coefficients instead of the d’Alembert equation should be considered, and the incident wave should be excited by a point source placed at a large but finite distance from the vortex instead of specifying an incident plane wave (which is not a solution of the governing equations).Here we use the new formulation of Belyaev and Kopiev to obtain the correct solution for the problem of non-resonant sound scattering, to second order in Mach number M. The partial harmonic expansion approach and the method of matched asymptotic expansions are employed. The scattered field in the region far outside the vortex is determined as the solution of the convective wave equation, and van Dyke's matching principle is used to match the fields inside and outside the vortical region. Finally, resonant scattering is also considered; an O(M²) result is found that unifies earlier solutions in the literature. These problems are considered for the first time.     

The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa-Holm, and generalized nonlinear Schrödinger equations

With the aid of the ordinary differential equation (ODE) involving an arbitrary positive power of dependent variable proposed by Li and Wang and an indirect F-function method very close to the F-expansion method, we solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l,n,p) and the generalized nonlinear Schrödinger equation with nonlinear dispersion GNLS(l,n,p,q). Taking advantage of the new subsidiary ODE, this F-function method is used to map the solutions of C(l,n,p) and GNLS(l,n,p,q) equations to those of that nonlinear ODE. As result, we can successfully obtain in a unified way, many exact solutions.     

Coherent Planar Symmetric Spacetimes Generated by Progressive Waves of Nambu–Goldstone Bosons

Within a SO(3,1) ?gauge invariant pseudo-orthonormal (Cartan) formalism, in the present paper, we are going to deal with the Einstein–Nambu–Goldstone system of equations, for a manifold with at least G₄ up to G₆ group of motion and a massless source-field excited along the z ?direction. This is also equivalent with the pure radiation energy–momentum tensor coming from circularly polarized waves generated by a rotating magnetic field. The corresponding essential equation which establishes the connection between the spacetime geometry and the matter-field is solved in some physically interesting cases.     

Stochastic theory of nonlinear rate processes with multiple stationary states

A comparison has been made between the deterministic and stochastic (master equation) formulation of nonlinear chemical rate processes with multiple stationary states. We have shown, via two specific examples of chemical reaction schemes, that the master equations have quasi-stationary state solutions which agree with the various initial condition dependent equilibrium solutions of the deterministic equations. The presence of fluctuations in the stochastic formulation leads to true equilibrium solutions, i.e. solutions which are independent of initial conditions as t → ∞. We show that the stochastic formulation leads to two distinct time scales for relaxation. The mean time for the reaction system to reach the quasi-stationary states from any initial state is of $\text{O}$(N⁰) where N is a measure of the size of the reaction system. The mean time for relaxation from a quasi-stationary state to the true equilibrium state is $\text{O}$(e^N). The results obtained from the stochastic formulation as regards the number and location of the quasi-stationary states are in complete agreement with the deterministic results.     

Symmetry-improved CJT effective action

The formalism introduced by Cornwall, Jackiw and Tomboulis (CJT) provides a systematic approach to consistently resumming non-perturbative effects in Quantum Thermal Field Theory. One major limitation of the CJT effective action is that its loopwise expansion introduces residual violations of possible global symmetries, thus giving rise to massive Goldstone bosons in the spontaneously broken phase of the theory. In this paper we develop a novel symmetry-improved CJT formalism for consistently encoding global symmetries in a loopwise expansion. In our formalism, the extremal solutions of the fields and propagators to a loopwise truncated CJT effective action are subject to additional constraints given by the Ward Identities due to global symmetries. By considering a simple O(2)$\mathbb{O}(2)$ scalar model, we show that, unlike other methods, our approach satisfies a number of important field-theoretic properties. In particular, we find that the Goldstone boson resulting from spontaneous symmetry breaking of O(2)$\mathbb{O}(2)$ is massless and the phase transition is a second-order one, already in the Hartree–Fock approximation. After taking the sunset diagrams into account, we show how our approach properly describes the threshold properties of the massless Goldstone boson and the Higgs particle in the loops. Finally, assuming minimal modifications to the Hartree–Fock approximated CJT effective action, we calculate the corresponding symmetry-improved CJT effective potential and discuss the conditions for its uniqueness for scalar-field values away from its minimum.     

Comparison between the generalized tanh–coth and the (<Emphasis Type="Italic">G</Emphasis>′/<Emphasis Type="Italic">G</Emphasis>)-expansion methods for solving NPDEs and NODEs

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G ^′/G )-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.     

Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods

A robust methodology is presented for efficiently solving partial differential equations using Chebyshev spectral techniques. It is well known that differential equations in one dimension can be solved efficiently with Chebyshev discretizations, O(N) operations for N unknowns, however this efficiency is lost in higher dimensions due to the coupling between modes. This paper presents the “quasi-inverse“ technique (QIT), which combines optimizations of one-dimensional spectral differentiation matrices with Kronecker matrix products to build efficient multi-dimensional operators. This strategy results in O(N^2D?1) operations for N^D unknowns, independent of the form of the differential operators. QIT is compared to the matrix diagonalization technique (MDT) of Haidvogel and Zang [D.B. Haidvogel, T. Zang, The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials, J. Comput. Phys. 30 (1979) 167–180] and Shen [J. Shen, Efficient spectral-Galerkin method. II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comp. 16 (1) (1995) 74–87]. While the cost for MDT and QIT are the same in two dimensions, there are significant differences. MDT utilizes an eigenvalue/eigenvector decomposition and can only be used for relatively simple differential equations. QIT is based upon intrinsic properties of the Chebyshev polynomials and is adaptable to linear PDEs with constant coefficients in simple domains. We present results for a standard suite of test problems, and discuss of the adaptability of QIT to more complicated problems.     

On some properties of multidimensional hyperbolic quasi-gasdynamic systems of equations

We study a multidimensional hyperbolic quasi-gasdynamic (HQGD) system of equations containing terms with a regularizing parameter τ > 0 and 2nd order space and time derivatives; the body force is taken into account. We transform it to a form close to the compressible Navier–Stokes system of equations. Then we derive the entropy balance equation and show that the entropy production is similar to the latter system plus a term of the order of O(τ²). We analyze an equation for the total entropy as well. We also show that the corresponding residuals in the HQGD equations with respect to the compressible Navier–Stokes ones are of the order of O(τ²) too. Finally we treat the simplified barotropic HQGD system of equations with the general state equation and the stationary potential body force and obtain the corresponding results for it.     

Microscopic description of collective states near the yrast line of nuclei with stable octupole deformation

Collective states near the yrast line in nuclei with stable octupole deformation are discussed in the framework of the random phase approximation (RPA) based on the cranking model. These vibrational states are characterized by the quantum number of generalized signature (eigenvalue of the operator S_x = PR_x^?1(π)). In the zero-octupole deformation limit the RPA equations of motion are reduced to the well-known ones characterized by both values of parity and signature, respectively. The connection of the translational and rotational symmetry of the model hamiltonian with the spurious solutions of the RPA equation of motion is discussed. Expressions for the reduced probabilities B(E1), B(E2) and B(E3) are obtained. These expressions confirm the conclusions of phenomenological models for the strong E1 and E3 intraband transitions in nuclei with stable octupole deformation.     

Orthogonal polynomial solutions of the Fokker-Planck equation

We have tabulated the form of the coefficientsg ₁(x) andg ₂(x) as well as the boundary values [a, b] of the Fokker-Planck equation $$\frac{{\partial P(x, t)}}{{\partial t}} = - \frac{\partial }{{\partial x}}[g_1 (x)P(x, t)] + \frac{{\partial ^2 }}{{\partial x^2 }}[g_2 (x)P(x, t)],a \leqslant x \leqslant b$$ for which the solution can be written as an eigenfunction expansion in the classical orthogonal polynomials. We also discuss the problem of finding solutions in terms of the discrete classical polynomials for the differential difference equations of stochastic processes.     

Dimensional regularisation and instantons

Dimensional regularisation is applied to the calculation of the quantum corrections to the instanton tunnelling amplitude in an SU(2) gauge theory. The principal feature is the introduction of an n-dimensional field configuration (a “quasi-instanton”), which generalises the O(5) invariance of the instanton and allows a coordinatisation of the function space of fields in its neighbourhood. This enables the functional integral measure to be factorised, with integrations over the translation and dilatation degrees of freedom being extracted. It is shown that a conformally invariant definition of orthogonality must be used in relation to the zero-mode eigenfunctions of the small oscillations expansion, irrespective of regularisation. An O(n + 1) covariant formalism is employed. An unconventional choice of gauge fixing term, which is not a perfect square, is made and is shown to allow the important freedom of calculating in a gauge specified by an arbitrary parameter α.