Soliton-like Structures Research Articles

Lubricated pipelining: stability of core–annular flow. Part 4. Ginzburg–Landau equations

Nonlinear stability of core-annular flow near points of the neutral curves at which perfect core-annular flow loses stability is studied using Ginzburg-Landau equations. Most of the core-annular flows are always unstable. Therefore the set of core-annular flows having critical Reynolds numbers is small, so that the set of flows for which our analysis applies is small. An efficient and accurate algorithm for computing all the coefficients of the Ginzburg-Landau equation is implemented. The nonlinear flows seen in the experiments do not appear to be modulations of monochromatic waves, and we see no evidence for soliton-like structures. We explore the bifurcation structure of finite-amplitude monochromatic waves at criticality. The bifurcation theory is consistent with observations in some of the flow cases to which it applies and is not inconsistent in the other cases.

Modulated, frequency-locked, and chaotic cross-waves

Measurements were made of the wave height of periodic, quasi-periodic, and chaotic parametrically forced cross-waves in a long rectangular channel. In general, three frequencies (and their harmonics) may be observed: the subharmonic frequency and two slow temporal modulations — a one-mode instability associated with streamwise variation and a sloshing motion associated with spanwise variation. Their interaction, as forcing frequency, f, and forcing amplitude, a, were varied, produced a pattern of Arnold tongues in which two or three frequencies were locked. The overall picture of frequency-locked and -unlocked regions is explained in terms of the Arnold tongues predicted by the circle-map theory describing weakly coupled oscillators. Some of the observed tongues are apparently folded by a subcritical bifurcation, with the tips of the tongues lying on the unstable manifold folded under the observed stable manifold. Near the intersection of the neutral stability curves for two adjacent modes, a standing wave localized on one side of the tank was observed in agreement with the coupled-mode analysis of Ayanle, Bernoff & Lichter (1990). At large cross-wave amplitudes, the spanwise wave structure apparently breaks up, because of modulational instability, into coherent soliton-like structures that propagate in the spanwise direction and are reflected by the sidewalls.

A nonlinear gauge-invariant field theory of elementary particles

The field theory of Cooperstock and Rosen (Int. J. Theor. Phys. 28, 423 (1989)), which modelled leptons, is enlarged to encompass the two known quark families. The singularity-free field theory constructs the particles in solitonlike structures via the nonlinear interaction of three scalar fields and the electromagnetic field. The theory predicts particle sizes that are consistent with experiment.

Generation of strong MHD Alfvénic turbulence.

Strong Alfv\'enic turbulence containing a number of solitonlike structures propagating at super-Alfv\'enic speeds is generated self-consistently and studied by means of computer simulation. A one-dimensional hybrid (kinetic ions, fluid electrons) code is used to investigate the nonlinear evolution of an electromagnetic ion-beam instability that generates low-frequency Alfv\'en-like waves. As the instability develops, the field-aligned hydromagnetic waves steepen, forming a soliton that bifurcates several times, leading to a fully turbulent state.

Solitonlike structures and their self-similarity in nonlinear Fibonacci multilayered media.

Switching from low to total transmission in a nonlinear Fibonacci multilayered medium is examined. It is found that this switching is due to the formation of solitonlike structures characterized by a ${\mathrm{sech}}^{2}$ intensity distribution, which results from a delicate interplay between dispersion and nonlinearity in the medium. These are shown to possess the remarkable properties of scaling and self-similarity.

Transient solitons in stimulated Raman scattering.

It is shown that permanent, solitonlike structures in which the pump retains a large fraction of its initial energy cannot exist for finite energy pulses. Ultimately, solitonlike structures propagate to the back end of the pulse and disappear. A method for propagating pump energy over long distances is proposed.

The method of contour dynamics for axisymmetric vortical structures with spatially bounded vorticity

The method of contour dynamics, used in /1/ to describe the interaction of plane vortices, is extended here to axisymmetric vortical flows. Numerical modelling of the evolution of the simplest contours is used as the basis for proposing a hypothesis of the existence of periodic, vortical soliton-like structures.

Many-parameter routes to optical turbulence.

The output of an externally pumped passive nonlinear-optical ring resonator is known to exhibit chaotic dynamics under variation of one or more external control parameters. The dynamics of the internal transverse laser beam profile is described, in the good-cavity limit, by an infinite-dimensional discrete-time map in function space. We find that the asymptotic dynamical behavior of the internal resonator field is in marked contrast to earlier plane-wave predictions. Instead, self-focusing, self-defocusing nonlinearities, and linear diffraction, combined with the pump and feedback of the ring resonator, give rise to strong spatial modulation of the transverse profile. For a self-focusing nonlinearity, spatially coherent robust transverse solitonlike structures persist even though the beam is undergoing temporally chaotic motion. Using attractor embedding techniques on our numerical solutions, we isolate distinct routes to optical turbulence under variation of a number of different control parameters. In particular, we provide examples of Ruelle-Takens-Newhouse sequences, intermittency, and a nongeneric bifurcation involving period doubling of invariant circles. A main result of the paper is that many few-dimensional attractors can coexist in different regions of the infinite-dimensional phase space at fixed values of the external control parameters. These attractors are accessed by varying the systems initial conditions and can, under variation of an appropriate control parameter, independently undergo transition to chaos. Interaction between neighboring attractors appears to be responsible for numerically observed departures from generic behavior. We note striking similarities between some of our numerically generated instability sequences and recent experimental observations in low-aspect-ratio fluid systems.

Nonlinear cylindrical or spherical converging ion-acoustic pulses

The Korteweg–de Vries equation which describes cylindrical or spherical converging ion-acoustic waves is solved numerically. Soliton-like structures are shown to be generated not only from compressive but also from rarefactive and mixed initial pulses. The velocity of the soliton-like structures is dependent on the initial pulse shape.

Strongly turbulent stabilization of electron beam-plasma interactions

The stabilization of electron beam interactions due to strongly turbulent nonlinearities is studied analytically and numerically for a wide range of plasma parameters. A fluid mode coupling code is described in which the effects of electron and ion Landau damping and linear growth due to the energetic electron beam are included in a phenomenological manner. Stabilization of the instability is found to occur when the amplitudes of the unstable modes exceed the threshold of the oscillating two-stream instability. The coordinate space structure of the turbulent spectrum which results clearly shows that soliton-like structures are formed by this process. Phenomenological models of both the initial stabilization and the asymptotic states are developed. Scaling laws between the beam-plasma growth rate and the fluctuations in the fields and plasma density are found in both cases, and shown to be in good agreement with the results of the simulation.

Soliton-like structures in plasmas

Abstract : Studies of the generation of soliton-like structures in plasmas lead to a class of nonlinear wave equations in which an external source plays a key role. Several related problems are briefly discussed with the hope of stimulating further work on this important topic by the applied mathematics community. (Author)