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We investigate the stability and dynamics of nonlinear structures that arise from the KadomtsevPetviashvili equation. For initially flat propagating cnoidal waves and solitons, the region of instab...

Much research has been done on the dynamics of the peptide chain. The amount of experimental, theoretical and numerical work on structure, stability and folding is staggering. However, there is no widely known analytical model derived from first principles. In this paper we derive equations for the dynamics of a peptide chain in a “quadrilateral chip” model, passing to the continuum from the discrete chain structure and assuming various local interactions. Ours is therefore a kinematic chain [1, 2]appearing in many mechanical structures. However, we include the appropriate constraints peculiar to peptides. Dihedral interactions are included and, when strong, are seen to ensure stability. Thus, they have the same effect as the non-local hydrogen bonds. The celebrated helices are exact particular solutions of our general equations. In order to obtain eigenfrequencies, a simplified model is introduced and peptide oscillations with frequencies of the order of 10 13 s −1 are obtained, in agreement with observations. Solitons are found in very restricted cases, thus far. External forces, such as those due to the solvent, are included in the model (Section 8).

This Letter reports on three-dimensional simulations that follow exact, {ital z} symmetric soliton solutions to an important equation of plasma physics and fluid dynamics. These solitons are seen to break up when perturbed slightly along {ital z}. Some fairly robust, three-dimensional entities are subsequently produced. However, they do not seem to resemble known, azimuthally symmetric solutions. An explanation of this interesting fact is offered. {copyright} {ital 1996 The American Physical Society.}

This Letter reports on numerical simulations showing exact, flat soliton solutions to a physical equation (that of Zakharov and Kuznetsov) breaking up into spherical solitons. Depending on the size of the box in which calculations are performed, the breakup can be either direct or via an intermediate, cylindrical soliton stage.

A simple dipole chain is the main building block of many physical models in solid state physics. Here, just such a chain of essentially identical dipoles is investigated by considering nearest-neighbor interactions of the individual charges in the dipoles. A three-dimensional model is found. In a restricted, two-dimensional picture, a nonlinear, second-order partial differential equation,more general than sine-Gordon, results. Soliton-kink solutions are found

Some problems connected with nonlinear wave instabilities are outlined, starting with the work of Zakharov, Benjamin, Feir and others in the sixties, up to recent work on the formation of cylindrical vortex entities from these waves. The Zakharov-Kuznetsov equation is seen to be a good testing ground and to admit analytical treatment where this would have been difficult for gravity waves, the original field of interest. An extension of this equation is suggested. It gives both Kadomtsev-Petviashvili and Zakharov-Kuznetsov in the two limits (zero and very strong magnetic field). In the second part of this survey, cylindrical and spherical ion acoustic, collapsing solitons are looked at, both by considering some solutions to model equations and by quoting results of a few recent experiments. Comments on current areas of research and unsolved problems wind up the article.

For pt.I see Ibid. vol.2, p.7143 (1990). A previous investigation of the stability of static one-dimensional solutions of the Landau-Ginzburg equation with a quartic non-linearity is extended. Exact spatially varying solutions are modified by small amplitude, time-dependent perturbations. In contradiction to the case of part I of this study, these are not assumed to have small frequency omega and small decay rates gamma . The authors show that all periodic solutions, as well as the solitary waves, are unstable with respect to this new type of perturbations. The kink solution is stable with respect to all perturbations considered. When the results of both parts of this paper are put together, they obtain an extensive stability analysis of static solutions to the Landau-Ginzburg equation. This equation is important in fluid dynamics, solid state and superconductivity theory, as well as other branches of physics. The paper is self-contained and can be read independently of part I.

A Comment on the Letter by K. Stasiewicz, Phys. Rev. Lett. 96, 175003 (2006). The authors of the Letter offer a Reply.

It has been known for some time that there is a hydrodynamic instability in rotating vortex-permeated superfluid HeII. This instability sets in at a critical relative velocity of the two components (normal and superfluid). The first theoretical treatment assumed a uniform constant velocity of the normal component, thus limiting the dynamics to the superfluid. Growth rates were found. A little later, it was shown that the critical velocity for the onset of the instability so obtained survived a more rigorous two-component calculation. However, growth rates obtained from the one fluid model are a different story and remain questionable (the original calculation is inconsistent with an expansion in the ratio of the two densities). Here the problem is solved by considering the two-component model. Generally speaking, growth rates are somewhat different from those hitherto used. This is important in view of recent experiments in which the superfluid matches the vorticity of the normal fluid.

This paper extends an investigation of the stability of intense optical slab beams in infinite plasmas [E. Infeld and G. Rowlands, Phys. Rev. A 42, 1005 (1990)]. These beams can be profiled in the shape of nonlinear waves or else solitons. In contradistinction to the earlier, purely theoretical investigation, perturbations of arbitrary wavelength are considered numerically. The results still imply radial stability, but also azimuthal instability of the soliton pulse. The wave structures, however, are unstable in both directions

Recently Frycz and Infeld performed a numerical calculation that yielded a stage-by-stage picture of a spontaneous transition from flat to an array of cylindrical solitons [Phys. Rev. Lett. 63, 384 (1989)]. In this Brief Report we look at some calculations that endeavor to explain both the collapse of a flat soliton, leading to cylindrical entities, and also the collapse of cylindrical solitons when instabilities along the axes are allowed

This paper investigates the stability of intense optical slab beams in infinite plasmas. Profiles depending on the distance x were found in a recent paper by Kurki-Suonio, Morrison, and Tajima [Phys. Rev. A 40, 3230 (1989)]. The most useful profiles were in the form of solitons. Solutions for which the beam profile was of oscillatory character were also found. Here a stability analysis is performed on all these profiles. The results, in terms of the physical cylindrical variables, imply radial stability but azimuthal instability of the solitons. The wave structures, however, are also unstable in the radial direction, and this explains numerical results of Kurki-Suonio et al.

Some time ago, we found an exact solution for nonlinear cold-plasma oscillations against a periodic ion background. The main and somewhat surprising physical effect due to the fixed ion background was the appearance of unbounded-electron density and electric-field intensity peaks. Along with others, we then tried to understand this in terms of realistic physical situations. Here we present a calculation indicating that these infinities can be removed by inclusion of viscous effects. A discussion of possible regions of validity of the model is given. For removal of infinities, the viscosity must be larger than a critical value depending on both the amplitude and spacing of the periodic ion background. Lagrangian coordinates are crucial in this calculation.

Nonlinear plasma oscillations against a background supporting an ion pulse or cavity are considered. We treat two physical cases. The first is that of a cold plasma, the second of a cold-ion--warm-electron plasma. The ion background includes a hyperbolic-secant-squared pulse or cavity but is otherwise uniform. An exact solution is obtained for the cold-plasma case. Infinite electric-field gradients result after some time, followed by a three-valued electric field, a physical impossibility. Thus some modification is called for. When the electrons are assumed to be thermal these infinite gradients can be limited. This will be the case if the ratio of pulse strength to background density is less than ${\ensuremath{\gamma}}^{1/3}$(${\ensuremath{\lambda}}_{\mathit{D}}$/l${)}^{2/3}$, where ${\ensuremath{\lambda}}_{\mathit{D}}$ is the Debye length and l a characteristic width of the pulse. Applications are briefly discussed.

Recently interest in nonlinear diffusion processes, such as the conductivity as a simple function of concentration, has increased. However, a symmetry of an important class of relevant nonlinear diffusion equations seems to have been missed in the published literature. This symmetry can be useful in constructing new solutions, the most important example corresponding to the diffusion of a \ensuremath{\delta}-shaped pulse located at the origin at time t=0. .AE

The dam breaking problem in terms of changing water levels behind and ahead of the dam, has been solved in the shallow water model. Here the effect of dam breaking on a symmetric downstream floating body is considered. A forced Duffing equation model, proposed by one of the authors, is used. The analysis has practical implications in terms of how far a marina should be situated from a large dam. Comparison with basin erosion, as caused by a water surface soliton, is instructive.

The results of a nonlinear stability numerical analysis for two- and three-dimensional solitons are presented. Soliton dynamics are described by the Kadomtsev-Petviashvili equation and by an equation suggested by one of us [E. Infeld, Postepy Fiz. 34, 215 (1983)]. The bending turns out to be the simplest perturbation of the system. The results of the simulation confirm stability in both cases.

Electron plasma waves in both isothermal and adiabatic electron gases are shown to be stable with respect to small amplitude, three-dimensional perturbations.

Nonlinear waves, solutions of the Zakharov–Kuznetsov (ZK) equation for a dilute plasma immersed in a strong magnetic field, are studied numerically. It is found that the most unstable mode rides with the carrier wave and leads to period doubling. Owing to the simplicity of the ZK equation, this and other phenomena, similar to those observed for gravity waves on a water surface, can be fully investigated. Analytical methods are also explored. An expansion in the carrier-wave amplitude leads to good estimates for growth rates, a result so far unobtainable for water waves. This paper can be read independently of parts 1 and 2, but a brief summary of all three parts is given at the end.

The modulation instability of small-amplitude ion-acoustic waves is investigated in three dimensions for TiTi = 0.

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).

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.

This paper investigates the general problem of stability of Bernstein—Greene— Kruskal type waves. By investigating perturbations perpendicular to the wave, we obtain a general sufficient condition for instability. This is then extended to the case of magnetized plasmas with a uniform magnetic field in the direction of the BGK wave. New perturbed modes, having no counterpart in linear theory, are also found. Various special cases are considered and previous, more particular results confirmed.

In this paper we consider the nonlinear development of the plasma Rayleigh-Taylor instability, also known as the Kruskal-Schwarzschild instability. Numerical calculations suggest that the plasma-vacuum interface, which initially takes the shape of an exponentially growing cosine function, will become multivalued after a while. Bubbles are seen to develop and to tend to at least partially detach themselves from the plasma. This paper gives a simple model of the process on the interface of a plasma layer with a vacuum. Profiles obtained are very similar to those found numerically by other authors. These multivalued profiles acquire a new symmetry in the semi-infinite plasma limit.

Large amplitude ion acoustic waves and solitons in two component plasmas are investigated for stability. The soliton solutions are found to be stable, while the nonlinear waves are always unstable, though for a significant range of parameters they are only unstable to fully three-dimensional perturbations. The results in one dimension are compared with those obtained from the K. –de V. equation, which gives stability for the nonlinear waves and solitons. Agreement is surprisingly good for Mach numbers less than about 1.5. A three-dimensional generalization of the K. –de V. equation is considered but this leads to stability for all nonlinear solutions and hence is not a good model for nonlinear waves. It is, however, reasonable in the soliton limit.

The general problem of the dynamics and stability of nonlinear ionacoustic waves and solitons in magnetized plasmas is important in ionospheric and laboratory applications. Here the geometrical-optics approximation solution is presented in the form of a quartic inω/k. Some growth rate and phase diagrams are also presented. The external magnetic field is seen to play a dual role: it strongly destabilizes the waves, and it introduces resonances not unlike those of the Bernstein modes. A perpendicular perturbation analysis is seen to be much too restricted.

Nonlinear electron plasma oscillations in a warm plasma are studied by use of Lagrangean coordinates. Without recourse to amplitude expansion, the electron density is obtained as an explicit function of space and time. In the zero-temperature limit, a restriction on the previously found cold-plasma solutions is found.

An exact solution for nonlinear, cold electron plasma oscillations has been known for some time. Here the authors prove the stability of this one-dimensional oscillation with respect to general angle but small amplitude perturbations.

The Zakharov-Kuznetsov equation describing Korteweg–de Vries waves and solitons in a strong, uniform magnetic field is rederived taking space stretching to be isotropic. This equation is then used to investigate nonlinear waves and solitons for long-wave instabilities. A solid angle of instability develops around the plane perpendicular to the magnetic field. For weakly nonlinear waves this angle is very narrow: widening as the amplitude of the nonlinear wave is increased. The soliton wave is unstable for all directions other than parallel to the field. Previous results of other authors, limited to solitons and perpendicular propagation are recovered. Calculations are illustrated by polar diagrams for the perturbations. Some broader implications are pointed out.

Envelope solitons are shown to be stable with respect to three-dimensional, long wavelength perturbations. Apparent contradictions with previous work of other authors are explained.

Three-dimensional generalizations of two different forms of the Boussinesq equation are derived. They are investigated for stability of slowly varying nonlinear wavetrains. The results obtained are then compared with the stability properties following from the full water wave equations. Agreement is found to be good forh0k0{h_0}{k_0}(depth times wavenumber) of order one. This is very satisfactory, as the Boussinesq equations are only supposed to be valid for smallh0k0{h_0}{k_0}. In particular, one version of the Boussinesq equation is found to yield instability with respect toone-dimensionalperturbations forh0k0>1.5{h_0}{k_0} > 1.5(as against 1.36 for the full equations). Finally, a similar comparison is performed for the three-dimensional Korteweg—de Vries equation.

Nonlinear waves and solitons satisfying the Zakharov-Kuznetsov equation for a dilute plasma immersed in a strong magnetic field are studied numerically. Growth rates of perpendicular instabilities, found theoretically in part 1, are confirmed and extended to arbitrary wavelengths of the perturbations (the calculations of part 1 were limited to long-wave perturbations). The effects of instabilities on nonlinear waves and solitons are illustrated graphically. Pre-vious, approximate results of other authors on the perpendicular growth rates for solitons are improved on. Similar results for perturbed nonlinear waves are presented. The effects of two-soliton collisions on instabilities are investigated. Rather surprisingly, we find that the growth of instabilities can be retarded by collisions. Instabilities can also be transferred from one soliton to another in a collision. This paper can be read independently of part 1.

In this note the problem of stability of two hot collisionless streams of charged particles is considered. The masses, charges, densities, and temperatures are arbitrary and the distribution functions are modelled by one ‘resonance’ function for each stream. The problem of stability is resolved by Nyquist diagrams, and, for the case of equal plasma frequencies, also by solving the dispersion relation in ω. A comparison with two Maxwellians on the one hand, and a two step function model on the other, is given. Step functions appear to be too crude for this problem.

In part 1 (Infeld 1972), it was shown that a nonlinear stationary wave in a cold plasma is marginally stable to all perturbations that are functions of the spatial direction parallel to the direction of propagation only. The stability analysis is extended here in part 2 to include general spatial variations. Within the limitations of the treatment, it is found that, for wave amplitudes up to maximal value (onset of trapping), the wave is marginally stable. Hence one would expect these large-amplitude cold plasma waves to exist in practice.The one-dimensional analysis of part 1 is also extended to include waves with relativistic velocities, and such waves are again found to be marginally stable.

This paper deals with the problem of classification of the two-stream instability for hot plasmas. The method is based on the criterion of Fainberg, Kurilko & Shapiro (1961). The basis of the analysis is the ‘resonance’ distribution, but a modified resonance distribution, leading to better results, is also investigated. The stability limit, classification into absolute and convective instabilities, as well as asymptotic growth rates are given for some chosen values of stream densities and temperatures, and arbitrary stream velocities. A method for finding the maximum temporal growth rates is also outlined.

This paper gives a general method for finding the line of marginal absolute instability and lines of equal asymptotic growth rates in parameter space for streams and a collisional plasma when the stability analysis for the same problem without plasma is known. The region of absolute instability is seen to be diminished whenever ν ≠ 0 (collision frequency ≠ 0). For ν = 0 the plasma background does not change the region of absolute instability, but growth rates are diminished. For this case and two cold streams the asymptotic growth rates are found analytically. Previous results are seen to be too narrow and singular to give an adequate picture of what is happening.

Gives some new information on the motion of charged particles in a Coulomb type field plus a homogeneous magnetic field. When more than one particle is introduced into this field we ignore the interaction between them.