Use Of Perturbation Expansion Research Articles

A tutorial on the non-linear progressive wave equation (NPE). Part 2. Derivation of the three-dimensional Cartesian version without use of perturbation expansions

The NPE is obtained by a novel method which avoids the use of perturbation expansions (McDonald et al., Appl. Acoust., 43 (1994) 59–67). The single variable NPE is derived here from the continuity equation by assuming an irrotational flow and employing partially linearised versions of the momentum and state equations. This derivation sheds new light on the range of conditions of applicability of the NPE.

A Phase-Plane Analysis of Bursting in the Three-Dimensional Bonhoeffer–van der Pol Equations

With the use of perturbation expansions and asymptotic matching, bursting in the three-dimensional Bonhoeffer–van der Pol equations is shown to be the result of the interaction of two oscillatory modes, one of small amplitude and the other of large amplitude. The large oscillations are similar to the relaxation oscillations found in the two-dimensional van der Pol system, but the small oscillations have not been fully understood before. This analysis also explains the transition in the two-dimensional system from stable equilibrium to relaxation oscillations: the intermediate stage between the two are the small oscillations.

Cellular automaton formulation of passive scalar dynamics

Cellular automata modeling of the advection of a passive scalar in a two-dimensional flow is examined in the context of discrete lattice kinetic theory. It is shown that if the passive scalar is represented by tagging or ‘‘coloring’’ automation particles a passive advection–diffusion equation emerges without use of perturbation expansions. For the specific case of the hydrodynamic lattice gas model of Frisch, Hasslacher, and Pomeau [Phys. Rev. Lett. 56, 1505 (1986)] the diffusion coefficient is calculated by perturbation.

Vertical slender jets with surface tension

The shape of a vertical slender jet of fluid falling steadily under the force of gravity is studied. The problem is formulated as a nonlinear free boundary-value problem for the potential. Surface-tension effects are included and studied. The use of perturbation expansions results in a system of equations that can be solved by an efficient numerical procedure. Computations were made for jets issuing from three different orifice shapes, which were an ellipse, a square, and an equilateral triangle. Computational results are presented illustrating the effects of different values for the surfacetension coefficient on the shape of the jet and the periodic nature of the cross-sectional shapes.

Vertical slender jets

The shape of a vertical slender jet of fluid falling steadily under the force of gravity is studied. The problem is formulated as a nonlinear free boundary-value problem for the potential. Surface tension effects are neglected. The use of perturbation expansions results in a system of equations that can be solved by an efficient numerical procedure. Computations were made for jets issuing from orifices in various shapes including an ellipse, a rectangle, and an equilateral triangle. Computational results are presented illustrating the propagation of discontinuities and the formation of thin sheets of fluid.

Single-mode dynamics of convective instabilities in a horizontal liquid layer

A theoretical description of the single-mode dynamics of both the Rayleigh-B\'enard instability (RBI) and the Soret-driven instability (SDI) is provided, starting from the general conservation equations and exploiting a technique similar to that used for the laser instability. A system of nonlinear dynamic equations is found to describe self-consistently both the RBI and the SDI. The single-mode steady-state solutions are derived without making use of perturbation expansions. A quantitative theoretical explanation is found to account for such recent experimental results as the saturation of the horizontally averaged concentration gradient at midheight in the SDI, the appearance of relaxation oscillations in RBI and SDI transients, and the dependence of the oscillation period on the temperature difference between the plates. The analogy with the laser suggests the possibility of observing giant pulses in the transient well above threshold, and of measuring pretransitional fluctuations of the mode which goes unstable. Connections with the Lorenz model are also discussed.

Many-particle green functions in the vector meson theories

Non-renormalizable interactions of massive vector mesons are examined without making use of perturbation expansions in the coupling constant. By discussing the characteristic example of the pseudovector meson-fermion coupling, it is shown that in the many-body Green functions exponentially divergent expressions occur. These exponential singularities can partially be removed from Green functions by choosing suitable canonical variables. The whole theory is, however, undefined in its original form. A possibility for defining Green functions is the analytic continuation with respect to the square of the coupling constant.

Derivation of Generalized Master Equations

We present a simplified derivation of a generalized master equation for the diagonal part of the occupation probability density. This derivation is valid for systems of arbitrary volume. It does not require the use of perturbation expansions nor the use of a diagonal singularity condition. In addition, a similar derivation is presented of a generalized master equation for the nondiagonal part of the occupation probability density. These equations become identical to the generalized master equations of Van Hove and Janner, respectively, if a perturbation expansion is made, if a diagonal singularity condition is assumed, and if the limit of infinite volume is taken.