Why many theories of shock waves are necessary: Kinetic functions,equivalent equations,and fourth-order models

We consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena. These models admit under-compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation. Building on earlier work by LeFloch and collaborators, we investigate the numerical approximation of these models by high-order finite difference schemes, and uncover several new features of the kinetic function associated with physically motivated second and third-order regularization terms, especially viscosity and capillarity terms.On one hand, the role of the equivalent equation associated with a finite difference scheme is discussed. We conjecture here and demonstrate numerically that the (numerical) kinetic function associated with a scheme approaches the (analytic) kinetic function associated with the given model – especially since its equivalent equation approaches the regularized model at a higher order. On the other hand, we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized Camassa–Holm model. Finally, we investigate to what extent a kinetic function can be associated with the equations of van der Waals fluids, whose flux-function admits two inflection points.     

Chaotic phase oscillation of a proton beam in a synchrotron

We investigate the chaotic phase oscillation of a proton beam in a cooler synchrotron. By using direct perturbation method, we construct the general solution of the 1st-order equation. It is demonstrated that the general solution is bounded under some initial and parameter conditions. From these conditions, we get a Melnikov function which predicts the existence of Smale-horseshoe chaos iff it has simple zeros. Our result under the 1st-order approximation is in good agreement with that in [H. Huang et al., Phys. Rev. E 48 (1993) 4678]. When the perturbation method is not suitable for the system, numerical simulation shows the system may present transient chaos before it goes into periodical oscillation; changing the damping parameter can result in or suppress stationary chaos.     

Vibrational intensities in methane

The absorption intensities of the two infra-red active vibrations in methane have been obtained from a perturbation calculation on the equilibrium wave functions derived in the preceding paper. The perturbation field is the change in the potential field due to the nuclei which results from moving the nuclei in the vibrational coordinate concerned, and a simplified form of second order perturbation theory, developed by Pople and Schofield, is used for the calculation. The main approximation involved is the neglect of f and higher harmonics in the spherical harmonic expansion of the nuclear field. The resulting dipole moment derivatives are approximately three times larger than the experimental values, but they show qualitative features and sign relationships which are significant.

The experimental intensity measurements are interpreted and discussed in relation to these results.     

Triangular approximation for Ising model and its application to Boltzmann machine

When we consider a problem in information processing, it is convenient to formulate the problem by using a random Ising model in statistical physics. However, a kind of computational difficulty arises in a case that the number of nodes becomes large. Hence approximation schemes such as a mean field approximation and a Bethe approximation have been used extensively for overcoming the difficulty. When frustration is essential in some problems, the Bethe approximation gives unfavorable results. In those problems, more advanced approximation schemes are needed beyond the Bethe approximation. In the present paper, we present explicitly the triangular approximation, which is the next approximation to the Bethe approximation. We apply the obtained approximation scheme to a Boltzmann machine in order to investigate the validity of the triangular approximation.     

Renormalization of N = 2, 4 Yang-Mills theories with soft breaking of an extended supersymmetry by N = 1 mass term

N = 2, 4 Yang-Mills theories with soft breaking of an extended supersymmetry by mass terms are considered. It is proved that for N = 4 there are no ultraviolet divergences in the mass renormalization constants to all orders of perturbation theory. For N = 2 our two-loop calculations show that the charge and mass renormalization constants contain only one-loop divergences and are the same in this order. It is shown by direct calculation that mass terms can acquire finite quantum corrections starting from the two-loop approximation. The renormalization scheme dependence of N = 4 renormalization group functions is investigated. We have found that unlike renormalization schemes with minimal subtractions of divergences other renormalization schemes give a nonzero β-function. At nonzero masses the β-function in MOM schemes is not zero even at the one-loop level. In the massless case β≠0 beginning from the two-loop approximation.     

Optical soliton perturbation with quadratic-cubic nonlinearity using a couple of strategic algorithms

In this work, we derive bright, dark and singular soliton solutions to quadratic-cubic nonlinear media with perturbation terms being present. We perform the modified simple and the trial equation algorithms to the considered model. In addition, periodic singular wave solutions will be constructed by the integration schemes.     

A perturbation theory for the Korteweg-De Vries equation

By writing the perturbed Korteweg-de Vries equation (1) in operator form (2), we derive equations which are a basis for a perturbation method. In particular, in the first approximation, we obtain from them equations describing the evolution of the soliton amplitude and velocity. The present theory may be extended, also, to other nonlinear evolution equations if they are solved, without perturbation, by the inverse-problem method.     

Variational field theoretic approach to relativistic meson-nucleon scattering

Non-perturbative polaron variational methods are applied, within the so-called particle or world-line representation of relativistic field theory, to study scattering in the context of the scalar Wick-Cutkosky model. Important features of the variational calculation are that it is a controlled approximation scheme valid for arbitrary coupling strengths, the Green functions have all the cuts and poles expected for the exact result at any order in perturbation theory and that the variational parameters are simultaneously sensitive to the infrared as well as the ultraviolet behaviour of the theory. We generalize the previously used quadratic trial action by allowing more freedom for off-shell propagation without a change in the on-shell variational equations and evaluate the scattering amplitude at first order in the variational scheme. Particular attention is paid to the s-channel scattering near threshold because here non-perturbative effects can be large. We check the unitarity of a our numerical calculation and find it greatly improved compared to perturbation theory and to the zeroth order variational results.     

Quantum tunneling: A general study in multi-dimensional potential barriers with and without dissipative coupling

Quantum tunneling is formulated in terms of the time evolution of a localized state and thus shown to be dependent upon the eigenspectrum of the system Hamiltonian. A number of exactly solvable models with local and non-local double-well potentials are discussed, and it is shown how, for local potentials, other solvable models can be generated by using Gelfand-Levitan and Darboux transformations. Tunneling in multi-dimensional potential barriers is investigated under semi-classical approximation by developing the method of asymptotic expansion of the wave function for large quantum numbers and the WKB approximation for separable systems. General expressions for the imaginary-time tunneling trajectory are obtained in both methods and specific applications are discussed. Approximation schemes for non-separable systems are also presented. A general study of dissipative multi-dimensional tunneling is carried out by using the Gisin equation, the Schrödinger-Langevin equation and the complex potential model. It is shown that, in general, different models of dissipation are not equivalent in the tunneling context. Using these models one can show (a) the existence of critical damping beyond which no tunneling can occur, (b) that tunneling trajectories are dependent on the damping constant and (c) that dissipation may stabilize the excited state rather than the ground state. Finally the tunneling time delay in one-dimensional systems for undamped and for dissipative systems is formulated in terms of the phase shift, and this has been used to show that the effect of damping on the time delay is ignorable.     

Quasi-Particle Theory of Alfven Soliton Interaction in Plasmas

The collision of solitons due to Alfven waves in plasmas is studied in this paper by the aid of quasi-particle theory. The suppression of the interaction of solitons, in presence of the perturbation terms, is acheived by means of this theory. The perturbation terms that are considered in this paper are nonlinear damping, finite conductivity and Landau damping. The numerical simulations support the theory that was developed. PACS Codes: 02.30.Ik, 02.30.Jr, 52.35.Sb.     

A numerical method for solving the Vlasov–Poisson equation based on the conservative IDO scheme

We have applied the conservative form of the Interpolated Differential Operator (IDO-CF) scheme in order to solve the Vlasov–Poisson equation, which is one of the multi-moment schemes. Through numerical tests of the nonlinear Landau damping and two-stream instability, we compared the present scheme with other schemes such as the Spline and CIP ones. We mainly investigated the conservation property of the L1-norm, energy, entropy and phase space area for each scheme, and demonstrated that the IDO-CF scheme is capable of performing stable long time scale simulation while maintaining high accuracy. The scheme is based on an Eulerian approach, and it can thus be directly used for Fokker–Planck, high dimensional Vlasov–Poisson and also guiding-center drift simulations, aiming at particular problems of plasma physics. The benchmark tests for such simulations have shown that the IDO-CF scheme is superior in keeping the conservation properties without causing serious phase error.     

Tilt-invariant theory of rough-surface scattering: I

The small-slope approximation (SSA) in rough-surface scattering theory uses the surface slope as a small parameter of expansion. But, from the physical point of view, the slope may not be a restrictive parameter because we can change the slope of a surface simply by tilting the coordinate system. We present the theory of rough-surface scattering in a coordinate-invariant form. The new method, tilt-invariant approximation (TIA), leads to a different expansion that does not require that the slope of a surface be small. For a small Rayleigh parameter this approximate solution provides the correct perturbation theory, for a large Rayleigh parameter it provides the Kirchhoff approximation with several correcting terms.     

Higher-order approximations for the particle-particle propagator

A new type of approximations for many-body Green's functions proposed recently is applied to the particle-particle (pp) propagator for anN-particle fermion system. The new approach which is referred to as the algebraic diagrammatic construction (ADC) is based on an exact resummation of the perturbation series for the pp-propagator in terms of a simple algebraic form introducing energy-independent effective interaction matrix elements and transition amplitudes. These effective quantities are represented by perturbation expansions and can be determined consistently through a given ordern of perturbation theory by comparing the algebraic form with the diagrammatic perturbation series of the pp-propagator through ordern. By this procedure one obtaines a systematic set of approximation schemes (ADC(n)) that represent infinite partial summations for the pp-propagator being complete throughnth order of perturbation theory. The explicit ADC equations forn=1 and 2 are presented and discussed. Comparison is made with the particle-particle random phase approximation (RPA). It is demonstrated that the second-order ADC scheme constitutes an essential step beyond the RPA which is consistent only through first order.     

Quantum decay rate of a nonlinear dissipative system with fission-like potential nearT c

We use a thermodynamic scheme (imaginary free energy method) in terms of the path integral technique to study the quantum decay rates of a metastable state system coupled to a heat bath in the crossover temperature (T _c) region. In this region the transition between thermally activated decay and tunneling occurs. A nonlinear coupling form factor is used to overcome the divergent integral in the partition function nearT _c. The decay rate formula based on the steepest descent approximation has been improved. A method is developed to calculate the real and imaginary parts of the partition function which combines a random walk method with fast-Fourier transform Monte-Carlo evaluation. For a nonlinear dissipative system with a damping correlation kernel of exponential form, the accurate numerical calculations are presented. The effects of nonlinear and frequency-dependent damping on the rate are shown.     

Spurious inertial oscillations in shallow-water models

For most of the discretization schemes, the numerical approximation of shallow-water models is a delicate problem. Indeed, the coupling between the momentum and the continuity equations usually leads to the appearance of spurious solutions and to anomalous dissipation/dispersion in the representation of the fast (Poincaré) and slow (Rossby) waves. In order to understand these difficulties and to select appropriate spatial discretization schemes, Fourier/dispersion analyses and the study of the null space of the associated discretized problems have proven beneficial. However, the cause of spurious oscillations and reduced convergence rates, that have been detected for most of mixed-order finite element shallow-water formulations, in simulating classical problems of geophysical fluid dynamics, is still an open question. The aim of the present study is to show that when spurious inertial solutions are present, they are mainly responsible for the aforementioned problems. Further, a criterion is found which determines the existence and the number of spurious inertial solutions. As it is delicate to cure spurious inertial modes, a class of possible discretization schemes is proposed, that is not affected by such spurious solutions.     

Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems

A unified approximation method is derived to illustrate the effect of electro-mechanical coupling on vibration-based energy harvesting systems caused by variations in damping ratio and excitation frequency of the mechanical subsystem. Vibrational energy harvesters are electro-mechanical systems that generate power from the ambient oscillations. Typically vibration-based energy harvesters employ a mechanical subsystem tuned to resonate with ambient oscillations. The piezoelectric or electromagnetic coupling mechanisms utilized in energy harvesters, transfers some energy from the mechanical subsystem and converts it to an electric energy. Recently the focus of energy harvesting community has shifted toward nonlinear energy harvesters that are less sensitive to the frequency of ambient vibrations. We consider the general class of hybrid energy harvesters that use both piezoelectric and electromagnetic energy harvesting mechanisms. Through using perturbation methods for low amplitude oscillations and numerical integration for large amplitude vibrations we establish a unified approximation method for linear, softly nonlinear, and bi-stable nonlinear energy harvesters. The method quantifies equivalent changes in damping and excitation frequency of the mechanical subsystem that resembles the backward coupling from energy harvesting. We investigate a novel nonlinear hybrid energy harvester as a case study of the proposed method. The approximation method is accurate, provides an intuitive explanation for backward coupling effects and in some cases reduces the computational efforts by an order of magnitude.     

The Post-NEWTONian Effect of Gravitation and the Retardation of the Gravitational Potential in Classical and Relativistic Theories of Gravitation

For planetary motions the post-NEWTON ian approximations of classical, special-relativistic, and general-covariant theories are compared. It is shown that, in this approximation, the anisotropy terms, which occur in the effective interaction potential in classical and special-relativistic theories, suggest a retardation of gravitation. In the post-NEWTON ian approximation of general-covariant theories the fixation of a retardation velocity is equivalent to coordinate conditions. – All post-NEWTON ian corrections are dipole-like ones, while, according to GAUSS , the classical perturbation theory generally leads to quadrupole-like corrections of the perturbation potential.     

Improved uncoupled Hartree-Fock perturbation theory

Various uncoupling schemes used in the first-order Hartree-Fock perturbation theory are compared. The analysis and extraction of the most important terms in the coupled Hartree-Fock perturbation scheme leads to a definition of the new functional for the determination of the first-order perturbed orbitals. This new functional represents an alternative uncoupling scheme for the first-order Hartree-Fock perturbation theory. Some special cases of real and pure imaginary perturbations and also the connections with previously proposed uncoupling schemes are discussed.

The uncoupling procedure proposed in this paper is illustrated by electric dipole polarizability calculations for some Be- and Ne-like atomic systems. The results obtained are almost as good as those calculated by using coupled Hartree-Fock perturbation theory.     

Quasinormal Modes of Spherically Symmetric Curve Spacetime with Dark Matter Term

We study quasinormal modes of scalar field perturbation and electromagnetic field perturbation in a black hole space-time with dark matter by using WKB approximation method. The result shows clearly that the real part of black hole quasinormal modes is mainly determined by angular quantum number while its imaginary part mainly determined by model number. We also found out that the dark matter will restrain the perturbation frequency and slow down the speed of damping in spacetime. In addition; dark matter has a greater influence upon quasinormal modes in the electromagnetic field than that in the scalar field.