The time dependent Ginzburg-Landau equation in fractal space-time

We use the hydrodynamic formulation of Scale Relativity Theory to analyze the TDGL equation. As a result, London equations come naturally from the system, when equating to zero the real velocity, the imaginary one turns real, the superconducting fluid act as a subquantum medium energy accumulator, the vector potential, the real and the imaginary velocity are all written in terms of the elliptic function. When solving the resulted system by means of WKBJ method, we get tunneling and quantization. In other words, scale transformation laws produce, on the motion equation of particles governed by the TDGL equation, under some peculiar assumptions, effects which are analogous to those of a “macroscopic quantum mechanics”.     

Granular matter and the time-dependent viscous eikonal equation

Deposition of granular matter under gravity can be described by the well-known two-layer model for a standing and a rolling layer. Matter from sources enters the rolling layer which flows along the gradient of the standing layer and finally enters the standing layer via interaction of the two layers. From this system of two coupled hyperbolic partial differential equations a time-dependent viscous eikonal equation is derived as a limiting case for weak sources, a thin rolling layer and fast convection of the rolling layer. This equation, supplied with boundary conditions, describes the deposition of dry sand from evenly distributed sources onto a flat table with a vertical rim of variable height. The stationary problem can also be seen as an application of the method of vanishing viscosity to the eikonal equation. For certain types of interaction between the two layers the resulting eikonal equation can be transformed into a linear equation. This transformation yields additional insight into the problem.     

The time dependent amplitude equation for the Swift-Hohenberg problem

Precise estimates for the validity of the amplitude approximation for the swift-Hohenberg equation are given, in a fully time dependent framework. It is shown that small solutions of orderO() which are modulated like stationary solutions have an evolution which is well described in the amplitude approximation for a time of orderO(^-2). For the proofs, we use techniques for nonlinear semigroups and oscillatory integrals.Dedicated to Res Jost and Arthur Wightman     

On the geometric structure underlying the eikonal equation

Given the eikonal equation _i=1 ³ (/x ⁱ ) ² =n ², we investigate the geometric structure that underlies the law of propagation of the wavefronts (x ¹,x ²,x ³) —ct=0. It turns out that Huygens' principle for the propagation of wavefronts is given in terms of a contact structure. Wavefronts are carried into wavefronts by contact transformations. As regards the wave-particle duality principle that arises in quantum mechanics, there is a natural geometric structure, a symplectic manifold (M ²ⁿ , ), which unifies Fermat's principle and the eikonal equation (Huygens' principle).On leave of absence from Institut für Angewandte Mathematik, Fachbereich Mathematik der Universität Mainz, Mainz, German Federal Republic.     

Fast sweeping method for the factored eikonal equation

We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part. The key idea is to enforce the causality of the original eikonal equation during the Gauss–Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation, i.e., the number of iterations for the Gauss–Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation, especially for point sources.     

A simple model for the time dependent Ginzburg Landau equation

A semiclassical model of the superconducting state, originally developped by de Gennes, is generalized for a time dependent energy gap. In this way a time dependent gap equation is obtained. By Taylor expansion one finds the time dependent Ginzburg-Landau-equation.     

Note on the convolutionless master equation with time dependent projector

In this communication we show that Fuliski-Kramarczyk master equation is valid for time-dependent projection operator. Thus, this equation is more general than the one proposed by Shibata and Hashitsume because no special assumption on the form of the projector is made.     

Momentum conservation,time dependent Ginzburg Landau equation and paraconductivity

Superconductors exhibit increasing electrical conductivity as the temperature approachesT _c from above, due to superconducting fluctuations. The functions σ_f1=σ(ω, ?)-σ _n (ω), ?=(T-T _c )/T _c , have been derived by Schmidt phenomenologically using the time dependent Ginzburg-Landau equation (TDGL). These functions fail to vanish in the absolute clean limit τ → ∞ as they must. We have therefore reinvestigated the derivation of the linearized TDGL-equation and the corresponding current expression in the presence of a time dependent vector potential. We find several new terms, which are important for the rather clean superconductor only and are easily interpreted physically in terms of momentum conservation. Applying these corrected equations to the paraconductivity problem, we derive σ_fl(ω, ?) which has an extra factor (1 —iωτ)^?2 compared to Schmidt's result. There is also an additional term, which is connected to the problem of the contribution calculated by Maki. By comparison with the linear response function belowT _c , we show that this term is valid in the limit ¦ω¦?¦Δ¦ only and may not be continued to ω=0. There remains, however, a problem connected with this term, which cannot be solved within the present phenomenological framework.     

A closed-form solution of the Schroedinger equation describing the harmonic oscillator with time dependent frequency and acted on by a time dependent perturbative force

A new form of the semiclassical quantum conditions in non-separable systems is proposed. In two dimensions (2D) it has the form (? = 1)

where C_Σ is the path of a classical trajectory closed in phase space, N_x and N_y are the number of circuits in the x and y ‘senses’ on the invariant toroid and J_x and J_y are the ‘good’ action variables on the toroid; these action variables, J_x and J_y , must have the values 2π(n ₁ + ½) and 2π(n ₂ + ½) respectively where n ₁ and n ₂ are the integer quantum numbers. Closed classical trajectories occur only for the exceptional toroids with rational frequency ratios. In the general case we imply that the trajectory has closed on itself to some arbitrary accuracy. Results for the 2D potentials studied are in agreement with previously published work. It is shown how the method may be extended to 3D systems.     

Knotted configurations with arbitrary Hopf index from the eikonal equation

The complex eikonal equation in (3 + 1) dimensions is investigated. It is shown that this equation generates many multi-knot configurations with an arbitrary value of the Hopf index. In general, these eikonal knots do not have the toroidal symmetry. For example, a solution with the topology of the trefoil knot is found. Moreover, we show that the eikonal knots provide an analytical framework in which qualitative (shape, topology) as well as quantitative (energy) features of the Faddeev-Niemi hopfions can be captured. This might suggest that the eikonal knots can be helpful in the construction of approximated (but analytical) knotted solutions of the Faddeev-Skyrme-Niemi model.Received: 5 April 2005, Revised: 16 May 2005, Published online: 19 July 2005     

Inverse scattering transform for the time dependent Schrödinger equation with applications to the KPI equation

For the direct-inverse scattering transform of the time dependent Schrödinger equation, rigorous results are obtained based on an opertor-triangular-factorization approach. By viewing the equation as a first order operator equation, similar results as for the first ordern x n matrix system are obtained. The nonlocal Riemann-Hilbert problem for inverse scattering is shown to have solution.     

Multidimensional nonlinear diffusion equation: Spatial time dependent diffusion coefficient and external forces

Despite many years of active research in the field and a number of fruitful applications, agent-based modeling has not yet made it through to the top ranking economic journals. In this paper we investigate why. We look at the following problematic areas: (i) interpretation of the simulation dynamics and generalization of the results, and (ii) estimation of the simulation model. We show that there exist solutions for both these issues. Along the way, we clarify some confounding differences in terminology between computer science and economic literature.     

Upwind monotonic interpolation methods for the solution of the time dependent radiative transfer equation

We describe the use of upwind monotonic interpolation methods for the solution of the time-dependent radiative transfer equation in both optically thin and thick media. These methods, originally developed to solve Eulerian advection problems in hydrodynamics, have the ability to propagate sharp features in the flow with very little numerical diffusion. We consider the implementation of both explicit and implicit versions of the method. The explicit version is able to keep radiation fronts resolved to only a few zones wide when higher order interpolation methods are used. Although traditional implementations of the implicit version suffer from large numerical diffusion, we describe an implicit method which considerably reduces this diffusion.     

Heating liquid dielectrics by time dependent fields

Steady state and time-resolved dielectric relaxation experiments are performed at high fields on viscous glycerol and the effects of energy absorption from the electric field are studied. Time resolution is obtained by a sinusoidal field whose amplitude is switched from a low to a high level and by recording voltage and current traces with an oscilloscope during this transition. Based on their distinct time and frequency dependences, three sources of modifying the dynamics and dielectric loss via an increase in the effective temperature can be distinguished: electrode temperature, real sample temperature, and configurational temperatures of the modes that absorbed the energy. Isothermal conditions that are desired for focusing on the configurational temperature changes (as in dielectric hole burning and related techniques) are maintained only for very thin samples and for moderate power levels. For high frequencies, say ν > 1 MHz, changes of the real temperature will exceed the effects of configurational temperatures in the case of macroscopic samples. Regarding microwave chemistry, heating via cell phone use, and related situations in which materials are subject to fields involving frequencies beyond the MHz regime, we conclude that changes in the configurational (or fictive) temperatures remain negligible compared with the increase of the real temperature. This simplifies the assessment of how time dependent electric fields modify the properties of materials.     

On the Schrödinger equation with a time dependent mass parameter: Nuclear fragmentation theory

We study the dynamics of a system with a coordinate-dependent inertia tensorB _ik (x ₁,...,x _n ), in which one of the coordinates is treated classically,x _j →x _j (t), so that the associated mass parameter becomes time-dependent. As a result, the Hamiltonian must contain a pure imaginary term in order that probability be conserved. The nuclear fragmentation theory is an example of such a system. The inertia parameter associated with the mass fragmentation degree of freedom depends on the relative motion coordinate and, hence, on time. The time dependence of the imaginary term is derived from a fully quantum mechanical treatment, by going to the classical limit in the relative motion variable. A connection is made with the closely related situation which arises if one transforms from an inertial system to one which depends non-linearly on the time.     

A time dependent Ginzburg-Landau equation and its application to the problem of resistivity in the mixed state

A time dependent modification of the Ginzburg-Landau equation is given which is based on the assumption that the functional derivative of the Ginzburg-Landau free energy expression with respect to the wave function is a generalized force in the sense of irreversible thermodynamics acting on the wave function. This equation implies an energy theorem, according to which the energy can be dissipated by i) production of Joule heat; ii) irreversible variation of the wave function. The theory is a limiting case of the BCS theory, and hence, it contains no adjustable parameters. The application of the modified equation to the problem of resistivity in the mixed state reveals satisfactory agreement between experiment and theory for reduced temperatures higher than 0.6.     

Faxén theorems for a spherically symmetric polymer in time dependent compressible flow

We derive Faxén theorems for the force, the torque and the symmetric force dipole moment acting on a spherically symmetric polymer suspended in an arbitrary time dependent compressible flow. The translational and rotational motions of the polymer are allowed to be non-steady as well. These results extend the theorems already derived for the case of a polymer immersed in a creeping flow.