Soliton-like waves

The result (173) applies to a photon model with the angular momentum h/2n of a boson, whereas the photon radius r would become half as large for the angular momentum h/4 of a fermion. Moreover, the present analysis on superposition of EMS normal modes is applicable not only to narrow linewidth wavepackets but also to a structure of short pulses and soliton-like waves. In these latter cases the radius in Eq. (173) is expected to be replaced by an average value resulting from a spectrum of broader linewidth. 

As stated above, their shape, their velocity, and their collisions characterize soliton-like waves. In the present state of knowledge, comparison between theoretical analysis and experimental data can be obtained in two basic situations when there is predominance of the nonlinearity dispersion balance, as in BKdV solitons (case A) and when the (free)-energy production dissipation terms dominate (case B). ... 

It also remains to be seen whether the present picture would allow for gravitational waves as predicted by the Einstein s general theory of relativity. On one hand, the present quantum model would not be against "action at a distance" on the other hand, a different, perhaps a soliton-like mechanism, would be needed to produce the latter. 

The field of Marangoni instabilities shows a large variety of dissipative structures, including the principle of stationary structures, hierarchical structures with limited self-similarity, relaxation oscillations and regular behavior of travelling autowaves with chaotic turbulence-like behaviour. There is also the oscillatory regime with trains of waves with soliton-like behaviour of each wave. Anormal as well as normal dispersion of these waves have recently... 

Pq the initial pressure, Qf the liquid density, and the initial void fraction), we find typically M = 5.0-5.5. This is approximately twice the speed of the shock wave. The detonation wave is a very stable system and it has a soliton-like structure. In Fig.2, a typical system of pressure profiles gives an impression of the evolution of the waves and the separation process. Unfortunately, due to the enormous scatter in the pressure, quantitative informa- tion on the wave profile cannot be obtained from a single pressure recording. Therefore, up to 50 profiles were superimposed in order to achieve a certain degree of smoothness in the profiles. 

An important characteristic of solitons is their non-dispersive (shape-conserving) motion. Conventional wave packets will lose their shape because the Fourier components of the packet propagate at different velocities. In a non-linear medium the velocity depends not only on the frequency of a wave but also on its amplitude. In favourable circumstances the effect of the amplitude dependence can compensate that of the frequency dependence, resulting in a stable solitary wave. A technical application of this idea is the propagation of soliton-like pulses in fibre optics, which considerably increases the bit rate in data transmission. 

Fig. 24. Space-time diagrams of solitary pulses constructed from PEEM images (a) wave splitting, (b) soliton-like behavior, (c) partial annihilation after [115].

I ig. 25. Simulation of solitary pulses. A defect with slightly higher oxygen sticking coefficient on the reconstructed surface is located around x = 40 /j,m. The same defect can explain wave splitting, soliton-like behavior and partial annihilation, (c) differs from (b) only in a stronger asymmetry of the initial conditions after [115]. 

Equation (6.335) is known to have an exact soliton-like solution, or solitary wave (also called a moving domain wall), when a/b < 1, and this solution was reported by Schiller, Pelzl and Demus [246] and Cladis and van Saar loos [43]. It is given by... 

Solitary waves, especially in shallow water, have been studied for many years[24]. They have the interesting property of interacting with other solitary waves and to separate afterwards as if there had been no interaction at all. This persistence of the wave led to the name soliton, to emphasize the particle-like character of these waves which seem to retain their identities in a collision. 

Many phenomena such as dislocations, electronic structures of polyacetylenes and other solids, Josephson junctions, spin dynamics and charge density waves in low-dimensional solids, fast ion conduction and phase transitions are being explained by invoking the concept of solitons. Solitons are exact analytical solutions of non-linear wave equations corresponding to bell-shaped or step-like changes in the variable (Ogurtani, 1983). They can move through a material with constant amplitude and velocity or remain stationary when two of them collide they are unmodified. The soliton concept has been employed in solid state chemistry to explain diverse phenomena. 

An important property is that the BKdV solitary waves have particle-like properties when colliding with each other as first shown by Zabusky and Kruskal, who showed that, upon collision, such solitary waves cross each other without apparent deformation, hence the name solitons. Moreover, depending on the angle before collision, they experience, at most, a displacement in their trajectories originating in a temporary change in wave velocity, which is called a phase shifts... 

The solution is solitonic waves, i.e. waves that do not spread or disperse like normal waves but instead retain their shape and size as shown in Fig. 36.14. 

The conduction process is not simply the spread of electron waves throughout the solid, as in a crystalline metal. Instead, the charge on the soliton jumps from one location to a neighbouring one under the influence of an electric field. This is similar to ionic conductivity, and this produces the Arrhenius-like behaviour. 

The unpaired electron spin in trans type polyacetylene (PA) plays an important role as a soliton for the conduction. The PA developed by Shirakawa [40] is a semiconductor, but changes to a conducting polymer by adding dopants. The distribution of the spin along the chain is symmetrical like a wave centering around the midpoint of the soliton. The neutral soliton of the excited state is not a conduction carrier. However, when dopants like As P5,12 and similar agents are introduced and abstract electrons from the solitons, the formed carbanium ion solitons are converted to conduction carriers. When dopants like Li, Na and similar substances add electrons to the solitons, carboanion solitons also change to conduction carriers. 

In polyparaphenylene, a soliton wave is not possible because the two phases, quinoid and aromatic, are not of the same energy, which excludes free motion. A double defect is possible though, a bipolaron. Such a defect represents a section of the quinoid stmcture (in the aromatic-like chain), at the end of which we have two unpaired electrons. The electrons, when paired with extra electrons from donor dopants or when removed by acceptor dopants, form a double ion (bipolaron), which may contribute to electric conductance. 

Before we plot our soliton, that is, the function we should examine the exponential terms in Equation (27.74). For g = 0.8, the first of them becomes where t is time in picoseconds. This means that it could make sense to keep those two terms in Equation (27.74) only for very small t, up to around 1 ps. However, even for so small values of f, the breather, like the one in Figure 27.12 or Figure 27.13, does not exist because the period of the carrier wave is longer than 2 ps. Therefore, only a pure soliton, that is, an envelope-type soliton, exists in the DNA chain and, neglecting... 

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