Unstable Media Research Articles

Media noise, nonlinear distortions, and thermal stability in high density recording

Low media noise and high thermal stability are required to achieve high density magnetic recording. Reduction of magnetic grain size is required in order to minimize transition jitter. However, very small grains become unstable and spontaneously reverse their magnetization due to thermal fluctuations. Future high density (>20 Gbit/in/sup 2/) recording systems are likely to operate in a media noise dominated environment and could exhibit a certain amount of signal decay. Recording performance degradation caused by media noise and thermal decay is evaluated. Density-dependent amplitude decay is caused by the presence of demagnetization fields and results in increasing level of nonlinear distortions and overwrite degradation. Magnetization decay is accompanied by changes in media noise in an unstable media as determined by the relative contributions of the transition and particulate noises. Experimental measurements suggest, that channel bit error rate degradation in an unstable medium is determined by signal and noise evolution, as well as by pattern-dependent distortions.

Variety of nonlinear wave-breaking

We discuss the formation of typical (structurally stable) singularities in nonlinear wave breaking in stable and unstable media. The wake wave-break due to the inhomogeneity of the Langmuir frequency is accompanied by electron injection into the acceleration phase. In a wake wave excited behind a finite width laser pulse, the wave-breaking mechanism involves the increase, with the distance behind the pulse, of the curvature of the wake front, followed by the self-intersection of the electron trajectories. In the long wavelength limit, the Weibel instability which leads to the generation of a strong magnetic field, the relativistic self focusing and the Rayleigh–Taylor instability of a thin plasma slab provide examples of a common behavior with the rarefaction wave-breaking in unstable media. We present a solution of the Cauchy problem that describes the evolution of nonlinear perturbations of the Rayleigh–Taylor instability in terms of analytical functions of a complex variable.

Self-similarity and nonlinear dynamics of thermally unstable media

We apply the theory of continuous group transformations to equations governing optically thin gaseous media in the isobaric limit in order to find the class of self-similar solutions admitted of such equations. Depending on the character of the diffusivity (linear or nonlinear) three types of self-similar variables are found generating (i) the well-known root-mean-square law of heat transfer, (ii) a power-law enhanced or de-pressed by nonlinearity of diffusivity or radiation losses, and (iii) progressive wave type of solutions, respectively. The constraints on the form of the source (i.e., cooling) function have been determined for each solution. These results have been applied to the study of the nonlinear evolution of a thermal instability of a plasma as an example of astrophysical interest. We find that the nonlinear phase of the thermal instability leads to the formation of conductive/cooling fronts in the medium if a thermally unstable equilibrium point does exist. The specific dependence of diffusivity on temperature was shown to determine the shape of cooling/conductive fronts, while radiative losses influence the temporal behavior of perturbations. The transition between different types of self-similar solutions are discussed along with their stability.

Example of shock waves in unstable media: the focusing nonlinear Schrödinger equation

Dissipationless shock waves in modulational unstable one-dimensional media are investigated for the simplest example of the integrable focusing nonlinear Schrödinger (NS) equation. Our approach is based on the construction of a special exact solution of the Whitham-NS system, which “partially saturates” the modulational instability.

Quasi-chaplygin unstable media

The authors construct exact (self-similar) solutions of the equations of motion for unstable media. In the long-wavelength approximation the solutions are modelled by the equations of motion for an ideal gas with negative compressibility.

Analysis of a Hamiltonian Amplitude Equation

The behaviour of a nonlinear Hamiltonian amplitude equation describing the modulation of plane waves in unstable media is analysed. In the defocusing case where the nonlinearity is destabilizing, solutions of the equation can blow up in finite time. In the focusing case, there is a family of stable solutions and there are indications of chaotic orbits away from these solutions.

Soliton phenomena in unstable media

As a canonical equation for soliton phenomena in unstable media, the unstable nonlinear Schrödinger (UNS) equation, i q x + q tt +2| q| 2 q=0 is proposed. The UNS equation is derived for two physical systems, Rayleigh-Taylor instability and electron beam plasma. It is shown that the initial value problem can be solved by the inverse scattering method. The exact N-soliton solution gives information on the soliton interaction: the faster (slower) soliton has negative (positive) position shift. It is thus concluded that soliton interaction in unstable media is attractive. Further, it is reported that the UNS equation explains a recent observation on propagations of localized structures in electron beam plasma.

The theory and applications of the unstable nonlinear Schrödinger equation

Recent works on the unstable nonlinear Schrödinger (UNS) equation, iq x + q n +2| q| 2 q=0, are reviewed. It is shown that initial value problems can be solved by the inverse scattering method. Exact N soliton solutions gives an information on the soliton interaction in unstable media. Namely, the position shifts due to the collisions have opposite signs compared with the conventional nonlinear Schrödinger equation, iq t + q xx + 2| q| 2 q = 0. As applications of the UNS equation, two physical systems, the Rayleigh-Taylor instability problem and electron beam plasma, are discussed. In both systems, the dispersion relation has a critical wave number below which frequency becomes a complex number. It is shown that near the critical wave number the wave amplitude obeys the UNS equation. It is concluded that the UNS equation is a canonical equation which describes nonlinear modulations of wave amplitude in unstable media.

Solitons in an Unstable Medium

Taking a model equation u t t - u x x - m 2 sin u =0, the behavior and the roles of solitons in unstable media are studied. The properties of solitons and their mutual collisions are discussed based on the exact solutions derived by the inverse scattering method. Furthermore, an initial boundary value problem of the model equation is studied numerically. It is concluded that creation of solitons serves to stabilize the system.

Formation of residual stresses in components made out of laminated and fibrous composites during the hardening process

The application of the theory of the viscoelasticity of unstable media to the calculation of the residual stresses enables one to reveal effects due to the change in the properties of a solidifying binder under different sets of operating conditions for the technological process. At the same time, it once again confirms that thermal shrinkage makes a significantly larger contribution to the magnitude of the residual stresses than chemical shrinkage. The absolute values of the stresses are in satisfactory agreement with results that were obtained using simpler calculational schemes.

Modulation for Nonlinear Wave in Dissipative or Unstable Media

Modulation of a nonlinear wave in a dissipative and dispersive medium is considered by the method of multiple scales. The slow variables for the amplitude are determined by the coupling between the nonlinearity of the envelope wave and the dissipative or dispersive effect. A perturbation theory is developed for a system of equations which, when linearized, has a plane wave solution with complex frequency of a small imaginary part. Governing equation for the amplitude becomes a type of generalized nonlinear Schrödinger equation in three dimensions. For spatially periodic case, there may be the case that small dissipation can make the wave grow depending on the initial amplitude. As an illustrative example of the general theory, modulation of the convective mode in a fluid layer heated from below is considered.

New results in the application to beam-plasma systems of the theory of pulse propagation in unstable media

New results in the application to beam-plasma systems of the theory of pulse propagation in unstable media

Instabilities: Convective versus Absolute

The response of a medium to a pulse disturbance is determined by the zeros of the dispersion function $D(k,\ensuremath{\omega})$ in the complex domain of both the wave number $k$ and the frequency $\ensuremath{\omega}$, and the asymptotic nature of the response near the origin depends on the velocity $V$ of the frame of reference of the observer. In particular, a crucial feature for an unstable medium is the behavior, as a function of $V$, of the position of the saddle points of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}(k)=\ensuremath{\omega}(k)\ensuremath{-}kV$, defined by the requirements $D(k,\ensuremath{\omega})=0, \frac{\ensuremath{\partial}D}{\ensuremath{\partial}k}+\frac{V\ensuremath{\partial}D}{\ensuremath{\partial}\ensuremath{\omega}}=0,$ since $\mathrm{Im}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}$ for suitably defined saddle points gives the asymptotic growth in frame $V$. Thus, if $\frac{{d}^{2}\ensuremath{\omega}}{d{k}^{2}}$ is taken along the contour $D=0$, the differential equations for the variation of a saddle point $\frac{\ensuremath{\delta}k}{\ensuremath{\delta}V}={(\frac{{d}^{2}\ensuremath{\omega}}{d{k}^{2}})}^{\ensuremath{-}1}, \frac{\ensuremath{\delta}\ensuremath{\omega}}{\ensuremath{\delta}V}=V{(\frac{{d}^{2}\ensuremath{\omega}}{d{k}^{2}})}^{\ensuremath{-}1}, \frac{\ensuremath{\delta}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}{\ensuremath{\delta}V}=\ensuremath{-}k$ (only two of which are independent) can be used to plot the asymptotic pulse shape. A new phenomenon of general interest to the theory of singularities of a function of two complex variables is described which, in the present context, is associated with the significance or insignificance of a given saddle point in contributing to instability, and with how this significance may change with $V$. This (mathematical) phenomenon may also be of relevance to elementary-particle theory through its inportance to the theory of the analytic structure of scattering amplitudes. The results of the theory are applied to certain examples of plasma physics, obtaining for these cases quantitative descriptions of the asymptotic shape of a pulse disturbance in unstable media.

SATURATED AND UNSATURATED FLOW EQUATION IN AN UNSTABLE POROUS MEDIUM

The experimental establishment is described of the Darcy formula, by measuring the discharge through a soil sample under a given difference in head. It was found to apply for a linear uniform flow into a chemically and physically stable porous medium. It was later extended to three dimensional flow. The flow of fluid through porous materials can be characterized by different motivating forces and by the resistance of the material to a relative movement. At the same time, measurements of such relative motion are not always possible especially in unstable media. A general formulation is presented of the relative flow equation into which one can introduce cases of saturated as well as unsaturated flow, Darcy's flow or any other flow of fluid in a porous medium.