Finite Spatial Interval Research Articles

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

On one-dimensional diffusion processes living in a bounded space interval

We prove that under some assumptions a one-dimensional Itô equation has a strong solution concentrated on a finite spatial interval, and the pathwise uniqueness holds.

An Inverse Convolution Method for Regular Parabolic Equations

In this paper, the problem of determining an unknown boundary control of a parabolic distributed parameter system, evolving over finite or infinite time, from incomplete, approximate interior temperature measurements is investigated. First, an exact solution is obtained for square integrable boundary controls associated with a general class of parabolic equations on a finite spatial interval under the assumption that the interior temperature is known for all times. Then, it is shown that this exact solution can be used to develop a stable algorithm for the numerical solution of this problem, without the introduction of standard regularization techniques. This algorithm assumes only knowledge of a finite, discrete set of approximate temperature readings. One advantage of this inversion process is the availability of a priori error bounds based on the measurement errors and frequency of sampling that are obtained in this paper. Another useful feature is that it encompasses boundary controls which are arbitrary linear combinations of surface temperature and flux. Numerical results are presented.

Dynamics of a nonautonomous sine-Gordon system modeling long Josephson junctions

We study the dynamics of an ac and dc driven sine-Gordon system extending over a finite spatial interval. We use this system to model fluxon oscillations in long (one-dimensional) Josephson junctions in the presence of an rf current. For three different values of the normalized spatial interval, or junction length, and a fixed value of the loss parameter the response of the junction is scanned over wide regions of the (rf drive amplitude and frequency) plane. In these regions chaotic intermittency, quasiperiodicity and isochronous phase locking are the most interesting dynamical features. We investigate the range of isochronous phase-locking of the soliton oscillations to the external ac drive in parameter space regions where experimental verification of our data can be considered.

On the persistence of unstable bump‐on‐tail electron velocity distributions in the Earth's foreshock

Further evidence for the persistence of bump‐on‐tail unstable reduced velocity distributions in the Earth's electron foreshock is presented. This persistence contradicts our understanding of quasi‐linear saturation of the bump‐on‐tail instability; the distributions should be stabilized through velocity space diffusion too quickly to allow an observation of their unstable form. A modified theory for the saturation of the bump‐on‐tail instability in the Earth's foreshock is proposed and examined using numerical simulation and quasi‐linear theoretical techniques. It is argued that the mechanism due to Filbert and Kellogg and to Cairns which is responsible for the creation of the bump‐on‐tail velocity distribution in the foreshock is still operative during the evolution of the bump‐on‐tail instability. The saturated state of the plasma must represent a balance between this creation mechanism and velocity space diffusion; the saturated state is not determined by velocity space diffusion alone. Thus the velocity distribution of the saturated state may still appear bump‐on‐tail unstable to standard linear analysis which does not take the creation mechanism into account. The bump‐on‐tail velocity distributions observed in the foreshock would then represent the state of the plasma after saturation of the bump‐on‐tail instability, not before. Numerical simulations of two types are discussed. In the first, the particle distribution function and the field are assumed periodic over a spatial interval of several tens or hundreds of Debye lengths along the local magnetic field. In the second, nonperiodic boundary conditions are imposed on the distribution function to model the spatial gradients which are responsible for the creation of the bump‐on‐tail unstable distribution. This mechanism for creating the unstable distribution continues to drive the particle distribution unstable during the evolution of the bump‐on‐tail instability. The saturated velocity distributions of the periodic and nonperiodic simulated plasmas are compared. At saturation the periodic velocity distribution exhibits plateau formation while the nonperiodic distribution does not. A new quasi‐linear theory for the space‐averaged velocity distribution function is constructed. This theory includes the effects of nonperiodic boundary conditions; it can be used to compare the evolution of periodic and nonperiodic bump‐on‐tail unstable particle distributions. In addition, this theory is based on the filtered Vlasov‐Maxwell system of equations; velocity space filamentation has been removed from the distribution function. It is shown that this theory is applicable on a finite spatial interval within which the field has been expressed in terms of a discrete Fourier mode expansion. Thus direct comparisons between theory and simulation are possible. Some predictions of this new quasi‐linear theory are compared favorably to the results of the bump‐on‐tail simulations.

On the Confinement and the Vacuum Degeneracy in the Schwinger Model

It is shown that multiple vacua are physical states in the Schwinger model formulated in a finite spatial interval by constructing explicitly an operator solution in which the fermionic degrees of freedom appear only in charge, chiral charge and operators. These are best illustrated by taking the massies's limit of an operator solution to the pre-Schwinger model which is represented in Hilbert space with the usual fermionic charge-sector. We also describe another operator solution to the pre-Schwinger model expressed in terms of boson fields only. It turns out that vacuum state is a non-normalizable eigenstate of the charge. § 1. Introduction and summary It is widely believed that the strong interaction can be described by the theory of quarks coupled to non-Abelian gauge fields and quarks and gauge fields may be permanently bound within hadrons and never appear as asymptotic particles. To implement these ideas, various theoretical approaches have been made. One possible way is to find model field theories, simple enough to solve, but nontrivial enough to make sense. It is Casher, Kogut and Susskind 1l who for the first time made use of the Schwinger model,) quantum electrodynamics with a massless spinor field in two dimensional space-time, as a model of quark confine­ ment. It is well known that the model is exactly solvable and the confinement of the spinor field occurs_2l,sl Nakanishi 4l gave an operator solution of the Schwinger model in terms of asymptotic bose fields alone to describe the confinement of the spinor field IJ! rigorously, while in all previous works operator solutions involved a massless free spinor field ¢. It is noteworthy that the solution of Nakanishi possesses the following unusual properties besides: Wightman functions are non­ vanishing regardless of the numbers of IJ!'s and IJ!*'s, gauge invariance of the first kind is spontaneously broken, chiral charge is not well-defined and the Lorentz transformation property of IJ! is not the standard one. Subsequently Nakanishi 5l found also that the solution of Lowenstein and Swieca3l to the Schwinger model can ,be reconstructed by multiplying his solution with spurion operators*) independent of asymptotic fields. In this case one can construct the so-called topological vacua *l Spurion operators were introduced for the first time by Lowenstein and Sweica in the an­ alysis of gauge-invariant algebra.

Optimal control of distributed parameter systems defined on a semi-infinite spatial interval

The optimal control problem with quadratic cost criteria is solved for the class of systems described by a scalar parabolic partial differential equation where the spatial differential operator is linear, is of arbitrary order, and is defined on the real line x ≥ 0. The control is applied at the boundary x = 0 and throughout the interior x > 0. The Hamilton-Jacobi equation for the problem is solved by applying elementary matrix methods to obtain an expression of Greens formula which involves the boundary control explicitly. The Riccati equation is formulated and, for the ease where the spatial operator is self-adjoint, the spectral representation of the Riccati equation is derived. Previous results have shown that the spectral representation of the Riccati equation for a self-adjoint differential operator on a finite spatial interval, possessing only a point spectrum, consists of an infinite set of bilinear ordinary differential equations. The new results presented here for the case of an operator on a se...