Solitary Wave Structures Research Articles

Solitary waves in nonlinear beam equations : stability, fission and fusion

We continue work by the second author and co-workers onsolitary wave solutions of nonlinear beam equations and their stabilityand interaction properties. The equations are partial differentialequations that are fourth-order in space and second-order in time.First, we highlight similarities between the intricate structure ofsolitary wave solutions for two different nonlinearities; apiecewise-linear term versus an exponential approximation to thisnonlinearity which was shown in earlier work to possess remarkablystable solitary waves. Second, we compare two different numericalmethods for solving the time dependent problem. One uses a fixed griddiscretization and the other a moving mesh method. We use these methodsto shed light on the nonlinear dynamics of the solitary waves. Earlywork has reported how even quite complex solitary waves appear stable,and that stable waves appear to interact like solitons. Here we show twofurther effects. The first effect is that large complex waves can, as aresult of roundoff error, spontaneously decompose into two simplerwaves, a process we call fission. The second is the fusion of twostable waves into another plus a small amount of radiation.

Exact Solitary Wave Solutions of Surface Waves in a Convecting Fluids**The project supported by National Natural Science Foundation of China and Natural Science Foundation of Zhejiang Province

In the standard Painlevé analysis, the singular manifold can be extended to many different forms because of its arbitrariness. Using some kinds of standard and nonstandard truncation approaches for the extended singular manifolds to a convecting fluid, we can obtain abundant solitary wave structures.

Surface equation of falling film flows with moderate Reynolds number and large but finite Weber number

Waves on a thin liquid layer falling down a solid wall, either vertical or inclined, are studied by means of a reduced equation. This equation is developed by the regularized long-wave expansion method, which is a combination of the Padé approximation and the long-wave expansion. Its numerical solutions are compared with the calculations of the full Navier–Stokes equation, simplified Navier–Stokes equation (the “boundary-layer” equation), and the traditional long-wave equations, as well as with experimental measurements. When the Reynolds number R is as small as unity, the present equation agrees with the Navier–Stokes equation and also with the traditional long-wave equations. For larger values of R, the traditional long-wave equations lose their validity and make a false prediction, while the present equation agrees with the Navier–Stokes equation, as long as the rescaled Reynolds number δ*=R/W1/3 does not exceed unity in the case of vertical films. Unlike the “boundary-layer” equation developed by previous researchers and expected to be valid at moderate and large Reynolds number, the present equation governs the surface evolution alone without postulating to resolve the velocity field. The structure of the present equation, however, has a correspondence to the depth-averaged equations, which facilitates discussing the physical mechanism of the wave dynamics. In particular, the physical origin of the instability mechanism and the wave suppression mechanism are discussed in terms of Whitham’s wave hierarchy theory. The balance of several physical effects such as drag, gravity, and inertia are also discussed in this connection. The analysis of the tail structure of permanent solitary waves predicts that the R dependence of its tail length λ exhibits two distinct regimes in the λ-R diagram. The second regime, which is not predicted by the traditional long-wave equations, arises when the inertia effect becomes dominant.

Discrete Auroral Arcs and Nonlinear Dispersive Field Line Resonances

Dispersive effects in field line resonances (FLRs) are discussed in the context of potential structures, parallel currents, and auroral density cavities observed by the FAST satellite. Our model includes the Earth's dipole magnetic field, and accounts for electron inertia, electron thermal pressure, finite ion gyroradius effects, and field aligned variations of the plasma density and ambient electron and ion temperatures. For realistic backgound parameters, we show that finite plasma temperature effects determine the dynamics of FLRs and that solitary wave structures evolve out of the resonance region, producing deep density cavities above the polar ionospheres. Results are shown to be in reasonable agreement with ground and satellite observations, with the exception of the magnitude of low altitude electric fields.

Abundant solitary wave structures of the nonlinear coupled scalar field equations

It is shown that there may be more abundant solitary wave structures of the nonlinear coupled scalar field than those of single scalar fields. In this paper, starting from a known simple example which is used in particle physics and condensed matter physics, we obtained various exact solitary wave and conoidal wave solutions by solving 4, 3, +3, 3+4, 6, 5 and models. Generally, from an arbitrary given single scalar field we may obtain a subset of solutions which are also solutions of the nonlinear coupled scalar fields.

Solitary wave structures in presence of nonisothermal ions in a dusty plasma

The nonlinear properties of solitary wave structures in the presence of non-isothermal ions, electrons, and charged dust grains are reported. The modification in the amplitude and the width of the solitary wave structures due to the inclusion of the effects of reflected ions and charged dust grains are investigated.

Self-induced transparency solitary waves in a doped nonlinear photonic band gap material

We derive the properties of self-induced transparency (SIT) solitary waves in a one-dimensional periodic structure doped uniformly with resonance two-level atoms. In our model, the electromagnetic field is treated classically and the dopant atoms are described quantum mechanically. The resulting solitary waves take the form of ultrashort (picosecond) laser pulses which propagate near the band edge of the nonlinear photonic band gap (PBG) material doped with rare-earth atoms such as erbium. Solitary wave formation involves the combined effects of group velocity dispersion (GVD), nonresonant Kerr nonlinearity, and resonant interaction with dopant atoms. We derive the general Maxwell-Bloch equations for a nonlinear PBG system and then demonstrate the existence of elementary solitary wave solutions for frequencies far outside the gap where GVD effects are negligible and for frequencies near the photonic band edge where GVD effects are crucial. We find two distinct new types of propagating SIT solitary wave pulses. Far from Bragg resonance, we recapture the usual McCall-Hahn soliton with hyperbolic secant profile when the nonlinear Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ However, when the host nonresonant Kerr coefficient is nonzero, we obtain the first new type of soliton. In this case, the optical soliton envelope function deviates from the hyperbolic secant profile and pulse propagation requires nontrivial phase modulation (chirping). We derive the dependence of the solitary wave structure on the Kerr coefficient ${\ensuremath{\chi}}^{(3)},$ the resonance impurity atom density, and the detuning of the average laser frequency from the atomic transition. When the laser frequency and the atomic transition frequencies are near the photonic band edge we obtain the second type of soliton. To illustrate the second type of soliton we consider two special cases. In the first case, GVD facilitates the propagation of an unchirped SIT-gap soliton moving at a velocity fixed by the material's parameters. The soliton structure changes dramatically as the laser frequency is tuned through the atomic resonance. In the second illustrative case we set the Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ In this case, the solution is a chirped pulse which arises from the balance between GVD and the resonance interaction with the dopant atoms. Finally, we show that under certain circumstances, these solitary wave solutions may persist even in the presence of (subpicosecond) dipolar dephasing of the dopant atoms and absorption losses of the host PBG material, provided that the system is incoherently pumped. These results may be relevant to the application of PBG materials as optical devices in fiber-optic networks.

Modification of solitary waves by third-harmonic generation

The effect of phase-matched third-harmonic generation on the structure and stability of spatial solitary waves is investigated. A power threshold for the existence of two-frequency spatial solitons is found, and the multistability of solitary waves in a Kerr medium owing to a higher-order nonlinear phase shift caused by cascaded third-order processes is revealed.

Velocity and turbulence structure of density currents and internal solitary waves: potential sediment transport and the formation of wave ripples in deep water

Laser Doppler anemometry (LDA) was used to measure the instantaneous downstream and vertical velocities in a series of simple and reflected saline density currents in a lock-exchange flume tank. All the currents were turbulent and subcritical. Mean downstream fluid velocities were in excess of the head velocity by up to 30%, and instantaneous velocities were greater by up to 50%. Turbulence intensities were highest within the head, and generally greatest in the middle part of the current, but did not correspond with the level of highest mean velocities. The maximum Reynolds stress also occurred within the head; large negative values were associated with shear along the upper boundary of the current. Peaks of turbulence, Reynolds stress and shear velocity occurred in association with the arrival of reflections. In large-scale turbidity currents, such reflections would be capable of re-entraining and resuspending sediment deposited by the forward current. Some reflections take the form of solitary waves within a residual flow with a velocity vector in the opposite direction. In nature, these could produce symmetrical ripples in environments below storm-wave base.

PSEUDORECURRENCE AND CHAOS OF CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION

Recurrence, pseudorecurrence, and chaotic solutions for a continuum Hamiltonian system in which there exist spatial patterns of solitary wave structures are investigated using the nonlinear Schrödinger equation (NSE) with cubic and quintic terms. The theoretical analyses indicate that there may exist Birkhoff's recurrence for the arbitrary parameter values. The numerical experiments show that there may be Fermi–Pasta–Ulam (FPU) recurrence, pseudorecurrence, and chaos when different initial conditions are chosen. The fact that the system energy is effectively shared by finite Fourier modes suggests that it may be possible to describe the continuum system in terms of some effective degrees of freedom.

Altitude distributions of upward flowing ion beams and solitary wave structures in the Viking data

Upward flowing ion beams are a typical feature observed in the auroral acceleration region. Observations show that a significant part of the upward field‐aligned acceleration of the ions takes place at altitudes between 5000 and 10,000 km. Another frequently observed feature in these regions are solitary waves (or weak double layers, as they are often called). These solitary structures have been observed practically only inside ion beams. In this letter we compare the occurrences of ion beams and solitary waves as a function of altitude. It is shown that there is a clear correlation between the occurrence probabilities of these two phenomena. This suggests that some of the acceleration is due to the solitary waves. Alternative interpretations are that the ion beams, accelerated by some other mechanism, are the cause of the SWs, or that these two phenomena are independent consequences of the same physical process.

Solitary waves on a two-dimensional lattice

In this paper we analyse the solitary wave structures observed in a two-dimensional lattice using the methods detailed in an earlier paper. In particular we provide a prediction of the form of anisotropy of plane waves that was first noted in numerical calculations. All methods describe the effect of anisotropy on the width of solitary waves, but only one of the methods analysed is capable of describing the effect of anisotropy on the height. A difference in two forms of continuum approximation is noted that does not occur in simpler lattices, i.e. in one-dimensional lattices the methods give the same results, whereas in the lattice studied here, one is shown to be more accurate than the other.

Secondary and tertiary excitation of three-dimensional patterns on a falling film

The primary instability of a falling film selectively amplifies two-dimensional noise down-stream over three-dimensional modes with transverse variation. If the initial three-dimensional noise is weak or if it has short wavelengths such that they are effectively damped by the capillary mechanism of the primary instability, our earlier study (Chang et al. 1993a) showed that the primary instability leads to a weakly nonlinear, nearly sinusoidal γ1 stationary wave which then undergoes a secondary transition to a strongly nonlinear γ2 wave with a solitary wave structure. We show here that the primary transition remains in the presence of significant three-dimensional noise but the secondary transition can be replaced by a selective excitation of oblique triad waves which can even include stable primary disturbances. The resulting secondary checkerboard pattern is associated with a subharmonic mode in the streamwise direction. If the initial transverse noise level is low, a secondary transition to a two-dimensional γ2 solitary wave is followed by a tertiary ‘phase instability’ dominated by transverse wave crest modulations.

Investigation of breakup of gas bubbles and its effect on the structure of moderate-intensity solitary pressure waves in a liquid with gas bubbles

Investigation of breakup of gas bubbles and its effect on the structure of moderate-intensity solitary pressure waves in a liquid with gas bubbles

Structure of narrow two-component solitary waves in hydrogen bonded chains

We present a new method for calculating the structure of narrow solitary waves in an extension of the original Antonchenko model for hydrogen bonded systems. Taking advantage of the properties of the topological kink in the proton sublattice, we obtain a solution in the oxygen sublattice that includes discreteness intrinsically. A comparison with the solution derived by numerical simulations shows that this approach is significantly better than the standard continuum approximation, particularly when the solitary wave speed approaches the speed of sound in the oxygen sublattice. The method determines also the frequencies and amplitudes of the wave radiated in the wake of a narrow oxygen kink in the same velocity range.

Stability of solitary structures in the nonlinear Schrödinger equation

The transverse stability of solitary wave structures to the equationiψt + ψxx + sψyy + p|ψ|2ψ = 0is studied for the four cases; (p = 1, s = 1), (p = 1, s = −1), (p = −1, s = −1) and (p = −1, s = 1). Solitary waves in the x-direction are envelopes ψ = √2 sech(x) exp (it), for p = 1, and they are kinks; ψ = √2 tanh(x) exp (– 2it), for p = −1. For s = 1 an "energy principle" can be invoked, and instability for the envelope occurs in the range 0 < k2 < 3, and for the kink in the range 0 < k2 < 1. Here k is the wavenumber of the perturbation in the y-direction. Growth rates are estimated very simply by means of a Rayleigh-Ritz method. For s = −1, we review existing results for the envelope. It is unstable in a range 0 < k2 ⩽ 1.17 where the upper limit must be determined numerically. For the kink, we find a continuum of unstable, non-local perturbations covering the whole range 0 < k2 < ∞. On both sides of the kink structure (|x| ≫ 1) these perturbations are oblique, purely growing plane waves.We conclude that both envelopes and kinks are transversely unstable for all signs of p and s.

On the structure of solitary waves in molecular chains

The structure of solitary waves in a quantum molecular chain, with the contribution of phonons with different wavenumbers taken into account, is investigated. The problem is solved on the basis of a non-linear non-local wave equation. It is shown that there exist two types of localised collective excitations which depend on the excitation velocity nu and the carrier wave phase velocity nu c. The criteria are considered of continuity and stability of excited bound multiparticle states and the estimation of their validity for the typical parameter values of a molecular chain is presented.

Intense electromagnetic solitary pulses in plasmas

The quasistatic slow plasma response to an intense electromagnetic wave is shown to result in a solitary wave structure of an arbitrary amplitude.

Nonlinear interaction of a powerful laser with an electron plasma

Interaction of a high-power circularly polarized electromagnetic wave with forced Langmuir oscillations is shown to result in solitary wave structure whose electron density profile has a depression at the center and humps on the sides.

A yang-mills plane wavevs. plane symmetry

The non-Abelian plane wave is treated in a space-time with plane symmetry. The Einstein-Yang-Mills equations are solved systematically and a solitary-wave structure is found for the planesymmetric geometry.