Stationary Localized Solutions Research Articles

Theory of magnetic electron drift modes

The nonlinear theory of two-dimensional magnetic electron drift modes is outlined. A Hamiltonian structure and integral invariants are identified. Spectral and statistical properties are considered and an energy spectrum with several similarity ranges is found. The inhomogeneity of the system is reflected by spectral anisotropy. Stationary localized vortex solutions are given. The linear stability of vortex street solutions with respect to the long wavelength perturbations is established. It is shown that nonlinear stability cannot be proven using Arnol'd's method.

Transition from stationary to traveling localized patterns in a two-dimensional reaction-diffusion system.

The bifurcation from localized stationary solutions to traveling patterns is studied in two-dimensional reaction-diffusion systems. In the case of local kinetics, which is of the Bonhoeffer--van der Pol type, the bifurcation point can be computed directly in terms of the stationary solution for arbitrary parameters. A stripelike pattern, which is the extension of a one-dimensional localized pattern into the second spatial dimension, is considered as a concrete example of such a pattern. The branching mode is computed for piecewise linear kinetics in a singular limit. The bifurcation is predominantly subcritical. Eventually the linear stability analysis of the branching traveling pattern is performed. Following the bifurcating branch, the last mode to become stable is a laterally wavy perturbation with an arbitrarily long wavelength.

On small stationary localized solutions for the generalized 1-D Swift-Hohenberg equation.

We prove the existence of small localized stationary solutions for the generalized Swift-Hohenberg equation and find under some assumption a part of a boundary of their existence in the parameter plane. The related stationary equation creates a reversible Hamiltonian system with two degrees of freedom that undergoes the Hamiltonian-Hopf bifurcation with an additional degeneracy. We investigate this bifurcation in a two-parameter unfolding by means of the sixth-order normal form for the related Hamiltonian. The region where no localized solutions exist has been pointed out as well. (c) 1995 American Institute of Physics.

Exact localized solutions in supersymmetric models of boson-fermion interactions

A method is given for constructing stationary localized solutions of classical equations of motion in two-dimensional space-time in models of boson-fermion interactions of general type. Applicability of the method is associated with invariance of the model with respect to supersymmetric transformations.

Stationary propagating magnetic electron vortices

The nonlinear theory of the magnetic electron mode is developed. It is shown that the model equations describing this mode has stationary localized solutions of the monopole vortex type. The vortices travel in the direction perpendicular to the inhomogeneities of the plasma with a velocity determined by the ratio between the temperature perturbation and the magnetic field of the vortex. For the vortex to be stationary and localized, this velocity must be larger than the phase velocity of the linear waves.

Stationary waves in a nonlinear periodic medium: Strong resonances and localized structures. II. The continuous model.

We study here the stationary waves in a nonlinear medium whose propagation constant is harmonically modulated in space. We recover most of the physical results obtained in the discrete model studied in the preceding paper (part I). However, the bifurcations associated with strong (Arnol'd) resonances exhibit some new features. The problem of ``quasi-integrability'' of the wave equation near the bifurcations receives special attention. Finally we give some comments on the observability of the stationary localized solutions.

Stability of localized stationary solutions

The present paper is devoted to the study of the stablity properties of localized stationary solutions of a nonlinear degenerate diffusion equation involving a nonlocally convective term. The equation, which is related to population dynamics, has various stationary solutions. The paper shows that the most fundamental stationary solution is stable in a sense. The asymptotic stability is also proved.

Symmetric and antisymmetric solutions for nonlinear wave modulations in plasmas

The existence of symmetric and antisymmetric stationary localized solutions for nonlinear amplitude-modulated Langmuir waves and circularly polarized electromagnetic waves in plasmas is considered. It is shown that the antisymmetric Langmuir solitons exist in the entire range of the Mach number, namely, 0<M<1, whereas antisymmetric electromagnetic solitons exist for all values of the Mach number. The parameter regions for the existence of different types of symmetric soliton solutions are obtained explicitly. It is found that both the density-hump and the density-dip electromagnetic solitons exist for subsonic as well as supersonic values of the Mach number.

On the existence of stationary multisolitons

It is shown that nonlinear wave equations possess an infinite (but countable) set of localized stationary solutions which may be considered as stationary bound states of “elementary” solitons. The corresponding criteria are suggested and supported experimentally and numerically.

Coupled localized electron-plasma waves and oscillatory ion-acoustic perturbations

The equations describing the coupling of high-frequency electrostatic waves with ion fluctuations allow stationary one-dimensional localized solutions which have not been reported previously. It is shown that these solutions follow from scaling laws different from those known for sonic Langmuir solitons, but similar to those used for static ion response. The present treatment generalizes the theory for subsonic Langmuir solitons by consistently including the second harmonic contribution in the low-frequency response. The physical importance of the new solutions is discussed.

Stationary localized solutions in nonlinear classical fields

The existence of stationary localized solutions for the Dirac field interacting with Maxwell and pseudoscalar fields is studied.

Stimulated compton scattering in a plasma: II. Asymptotic behaviour-Bose condensations-onset of a final equilibrium state

We develop further our previous study of the self-consistent kinetics of a closed system constituted by interacting gases of electrons and photons. We point out the existence of a low frequency region where the evolution of the radiation spectrum always leads to a Bose condensation. We describe this condensation. Then we study the asymptotic behaviour of the interaction, where the linear induced term in the photon kinetic equation (related to T e) becomes dominant. We point out a “blue diffusion”, which leads finally to “local stationary solutions” in the high frequency region: then a sequence of Bose Einstein distributions with a negative chemical potential, defined on high frequency intervals and identified with these local solutions, is believed to lead to the final equilibrium state of the system. We point out the interest of such “negative μ-B.E.D.” in UV or X-ray astronomy.