On the degenerated soft-mode instability

We consider instabilities of a single mode with finite wavenumber in inversion symmetric spatially one dimensional systems, where the character of the bifurcation changes from sub- to supercritical behaviour. Starting from a general equation of motion the full amplitude equation is derived systematically and formulas for the dependence of the coefficients on the system parameters are obtained. We emphasise the importance of nonlinear derivative terms in the amplitude equation for the behaviour in the vicinity of the bifurcation point. Especially the numerical values of the corresponding coefficients determine the region of coexistence between the stable trivial solution and stable spatially periodic patterns. Our approach clearly shows that similar considerations fail for the case of oscillatory instabilities.Comment: 16 pages, uses iop style files, manuscript also available at ftp://athene.fkp.physik.th-darmstadt.de/pub/publications/wolfram/jpa_97/ or at http://athene.fkp.physik.th-darmstadt.de/public/wolfram_publ.html. J. Phys. A in pres

A Path Intergal Approach to Current

Discontinuous initial wave functions or wave functions with discontintuous derivative and with bounded support arise in a natural way in various situations in physics, in particular in measurement theory. The propagation of such initial wave functions is not well described by the Schr\"odinger current which vanishes on the boundary of the support of the wave function. This propagation gives rise to a uni-directional current at the boundary of the support. We use path integrals to define current and uni-directional current and give a direct derivation of the expression for current from the path integral formulation for both diffusion and quantum mechanics. Furthermore, we give an explicit asymptotic expression for the short time propagation of initial wave function with compact support for both the cases of discontinuous derivative and discontinuous wave function. We show that in the former case the probability propagated across the boundary of the support in time $\Delta t$ is $O(\Delta t^{3/2})$ and the initial uni-directional current is $O(\Delta t^{1/2})$. This recovers the Zeno effect for continuous detection of a particle in a given domain. For the latter case the probability propagated across the boundary of the support in time $\Delta t$ is $O(\Delta t^{1/2})$ and the initial uni-directional current is $O(\Delta t^{-1/2})$. This is an anti-Zeno effect. However, the probability propagated across a point located at a finite distance from the boundary of the support is $O(\Delta t)$. This gives a decay law.Comment: 17 pages, Late

Dispersion in time and space: what propagating optical pulses in time (& not space) forces us to confront

I derive a temporally propagated uni-directional optical pulse equation valid in the few cycle limit. Temporal propagation is advantageous because it naturally preserves causality, unlike the competing spatially propagated models. The exact coupled bi-directional equations that this approach generates can be efficiently approximated down to a uni-directional form in cases where an optical pulse changes little over one optical cycle. They also permit a direct term-to-term comparison of the exact bi-directional theory with its corresponding approximate uni-directional theory. Notably, temporal propagation handles dispersion in a different way, and this difference serves to highlight existing approximations inherent in spatially propagated treatments of dispersion. Accordingly, I emphasise the need for future work in clarifying the limitations of the dispersion conversion required by these types of approaches; since the only alternative in the few cycle limit may be to resort to the much more computationally intensive full Maxwell equation solvers.Comment: v3: updates and clarifications. arXiv admin note: text overlap with arXiv:0810.568

Equations of electromagnetism in some special anisotropic spaces

We show that anisotropy of the space naturally leads to new terms in the expression of Lorentz force, as well as in the expressions of currents.Comment: 15 page

Higher order dilaton gravity: brane equations of motion in the covariant formulation

Dilaton gravity with general brane localized interactions is investigated. Models with corrections up to arbitrary order in field derivatives are considered. Effective gravitational equations of motion at the brane are derived in the covariant approach. Dependence of such brane equations on the bulk quantities is discussed. It is shown that the number of the bulk independent brane equations of motion depends strongly on the symmetries assumed for the model and for the background. Examples with two and four derivatives of the fields are presented in more detail.Comment: 32 pages, references added, discussion extended, typos corrected, version to be publishe