Stability Of Nonlinear Waves Research Articles

On the stability of thermomagnetic waves in type II superconductors

The thermomagnetic instability of the critical state insuperconductors is analysed with an account of the dissipation and dispersion. Thepossibility of the existence of a nonlinear shock wave describingthe final stage of the instability evolution in a superconductor is demonstrated. The problem of nonlinear wave stability with respect to small thermal andelectromagnetic perturbations is studied. It it also shown thatonly damped perturbations correspond to the space-limited solutions, whichmeans that the nonlinear thermomagnetic wave is stable.

Stability of waves in discrete systems

The Evans function is a complex analytic function whose zeros correspondto eigenvalues of an associated linear operator. Over the last decade, it hasbeen shown to be a very powerful tool for examining eigenvalue problemsof linearized partial differential equations. In this work the Evansfunction is constructed for differential-difference, i.e. lattice,equations. It is then used to study the stability of waves inHamiltonian lattice models. In particular, the stability ofnonlinear waves for the discrete nonlinear Schrödingerequation, as well as for the discrete sine-Gordon equation, isestablished. The linear spectrum for each problem is identified,and terms which are exponentially small in the discretenessparameter are quantified. The analytic results are supplementedwith detailed numerical simulations.

On the stability of TE0waves guided by thin films bounded by a nonlinear medium

The authors show that investigations on the stability of nonlinear waves guided by planar structures greatly benefit when the conditions of marginal stability are considered on a suitable restricted parameter space. Relevant information is summarized in a few diagrams, so that waveguiding properties can be immediately read off from them, and critical power-independent values of the waveguide parameters to ensure stable TE0 propagation are obtained for a particular structure.

A numerical investigation of the nonlinear wave stability of vertical thermal boundary layer flow in a porous medium

A two-dimensional, nonlinear, time-dependent, elliptic, numerical method coupled with an appropriate coordinate transformation is used to investigate the stability of free convection induced by an isothermally heated semi-infinite surface embedded in a fluid-saturated porous medium. It is found that the basic boundary layer flow is stable even to large amplitude disturbance for nondimensional distances of up to 1024 from the leading edge of the heated surface.

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.

On the stability of plane wave and soliton solutions to the Zakharov equations

Plane wave and soliton solutions of the two types of Zakharov equation (two dimensional and simplified one directional) are considered. Stability properties in one dimensional space are seen to be similar. This is interesting, as the first type of equation is not solvable whereas the second is. The soliton solutions of both are one dimensionally stable but those of the full Zakharov equations are unstable with respect to perpendicular perturbations. Regions of stability of nonlinear wave and shock wave solutions in parameter space as well as growth rates of instabilities are given.

On the stability of nonlinear waves in integrable models

A new method of stability investigation is presented for solutions of nonlinear equations integrable with the help of the inverse scattering transform (IST). The stability problem for periodic nonlinear waves in weakly dispersive media is solved with respect to transverse perturbations. It is shown that for positive dispersion media one-dimensional waves are unstable, and for negative dispersion such waves are stable.

A simple derivation of the Mel'nikov condition for the appearance of homoclinic points

Techniques introduced in study of the stability of non-linear waves are used to give a simple derivation of the Mel'nikov condition for the appearance of homoclinic points in dynamical systems. The interpretation of this result in the context of perturbation theory is discussed.

Stability of nonlinear waves in hyperelastic multilayer media

Stability of nonlinear waves in hyperelastic multilayer media

Two-dimensional stability of large amplitude langmuir waves and solitons

The two-dimensional stability of nonlinear wave and soliton solutions of the exponentially nonlinear Schrödinger equation is examined. All stationary entities are unstable to two-dimensional perturbations. It is found that the saturable nonlinearity decreases growth rates in comparison with the small amplitude limit.

Propagation and stability of non-linear waves in spinning anisotropic membrane-like disks

The system of two non-linear coupled differential equations of the second and fourth orders, respectively, governing motions of a rotating anisotropic membrane-like disk, is analyzed under the assumption of a particular form of the transverse deflections. The general equations derived for the stresses are compared with the known particular solutions associated with both a non-linear purely vibrational mode and non-linear (and linear) wave propagation in anisotropic and isotropic materials. The stability of motion for a specific mode and a particular choice of temporal perturbation is investigated. General equations obtained for the perturbation parameters are examined in two particular cases.

Three dimensional stability of solutions of the nonlinear Schr�dinger equation

The three dimensional stability of nonlinear wave, soliton, and shock solutions of the nonlinear Schrodinger equation is examined. For the nonlinear waves the analysis is rigorous, whereas solitons are treated as limiting cases of the nonlinear waves. As two different limits yield the same result, this result is included. Shock solutions are again examined as limiting cases of the nonlinear waves, and for them an independent consideration of the limits at plus and minus infinity gives confirmation of the result. In contradistinction to the one dimensional analysis, which gave stability for some of the waves, the soliton and the shock, all these entities are now found to be unstable (though with varying degree of rigour in the treatment).

On the stability of nonlinear waves coexisting with plasma beams

In this paper we consider the stability of one-dimensional stationary waves set up by two counter-streaming beams of electrons in a background of stationary ions. The perturbations considered are long-wave in a direction perpendicular to the wave. The presence of a uniform magnetic field in the direction of the wave and the effect of a perpendicular pressure are taken into account. In the long-wavelength limit growth rates are diminished by the nonlinear wave. When the amplitude of this wave tends to its maximum value, the growth rates tend to zero. Thus the wave has a stabilizing effect for long-wave perturbations. Three- dimensional effects lead to additional instabilities which are also quenched by the nonlinear wave, but not as fast as the one-dimensional calculation indicates.

Nonlinear Wave Stability with Collisionless Shock Application

A detailed study of the stability of nonlinear waves in a collisionless plasma has been made. In addition to the ordinary two-stream instability, it has been found that another mode, the cross-stream instability, also plays an important role. The heating of electrons by two-stream instabilities can preclude the existence of nonlinear waves. The cross-stream instability is partly hydromagnetic; it may severely affect the ions and can cause macroscopic breakup of the nonlinear wave. For strong magnetic fields and low temperatures, it is found that the cross-stream instability does not destroy sufficiently weak nonlinear waves. The results are used to construct a collisionless shock wave.

A criterion for nonlinear wave stability

Nonlinear resonant wave triads composed of one finite and two infinitesimal components are unstable for sum interactions and neutrally stable for difference interactions. A similar criterion holds for tertiary interactions.