Solitary waves

This balance between nonlinearity and dissipation gives rise to what is called a solitary wave, and is correctly described by the Korteweg-de Vries (KdV) equation... 

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. 

Two lower states of the frans-(CH) are energetically degenerated as follows from symmetry conditions. Theory shows that electron excitation invariably includes the lattice distortion leading to polaron or soliton formations. If polarons have analogs in the three dimensional (3D) semiconductors, the solitons are nonlinear excited states inherent only to ID systems. This excitation may travel as a solitary wave without dissipation of the energy. So the 1-D lattice defines the electronic properties of the polyacetylene and polyconjugated polymers. 

A soliton is a solitary wave that preserves its shape and speed in a collision with another solitary wave [12,13]. Soliton solutions to differential equations require complete integrability and integrable systems conserve geometric features related to symmetry. Unlike the equations of motion for conventional Maxwell theory, which are solutions of U(l) symmetry systems, solitons are solutions of SU(2) symmetry systems. These notions of group symmetry are more fundamental than differential equation descriptions. Therefore, although a complete exposition is beyond the scope of the present review, we develop some basic concepts in order to place differential equation descriptions within the context of group theory. 

The simplest solution to (7.69), is a soliton, i.e., a solitary wave with unchanged profile, which runs at velocity u ... 

Consider a solitary wave, progressing to the left in an open channel, with celerity (wave velocity) c. We may replace this situation with the equivalent steady-flow case in which the wave stands still while the flow enters at velocity Vi = -c. Writing the energy equation, Eq. (10.122), between points 1 and 2 (with Z = z2, oq = a2 = 1, and neglecting friction), and keeping the same variable definitions, we have... 

Modeling EM solitary waves in a plasma is quite a challenging problem due to the intrinsic nonlinearity of these objects. Most of the theories have been developed for one-dimensional quasi-stationary EM energy distributions, which represent the asymptotic equilibrium states that are achieved by the radiation-plasma system after long interaction times. The analytical modeling of the phase of formation of an EM soliton, which we qualitatively described in the previous section, is still an open problem. What are usually called solitons are asymptotic quasi-stationary solutions of the Maxwell equations that is, the amplitude of the associated vector potential is either an harmonic function of time (for example, for linear polarization) or it is a constant (circular polarization). Let s briefly review the theory of one-dimensional RES. 

We can summarize the main results from the previous investigations in the following points (i) soliton solutions have been found under general conditions for an electron-positron plasma and by assuming quasi-neutrality in an electron-ion plasma (ii) sub-cycle nondrifting solitary waves represent an equilibrium in a multicomponent warm plasma that is, half-wavelengths of the EM radiation can be trapped inside a plasma density well (iii) the... 

A soliton is a giant solitary wave produced in canals by a cancellation of nonlinear and dispersive effects. The connection between aqueous solitons and tsunamis ("harbor waves") is not definitively established. In "doped" conducting polyacetylene, a neutral soliton is a collective excitation of a polyacetylene oligomer that has amplitude for several adjacent sites [57],... 

Soliton — Solitons (solitary waves) are neutral or charged quasiparticles which were introduced in solid state physics in order to describe the electron-phonon coupling. In one-dimensional chainlike structures there is a strong coupling of the electronic states to conformational excitations (solitons), therefore, the concept of soliton (-> polaron, - bipolaron) became an essential tool to explain the behavior of - conducting polymers. While in traditional three-dimensional -> semiconductors due to their rigid structure the conventional concept of - electrons and -> holes as dominant excitations is considered, in the case of polymers the dominant electronic excitations are inherently coupled to chain distortions [i]. 

An extra electron put on a polymer chain deforms the chain and forms a SWAP (Solitary Wave Acoustic Polaron). The SWAP dynamics and energy dissipation are such that the smallest field causes it to move at approximately the sound velocity its mobility is ultra high, higher than that of any conventional semiconductor. Increasing field changes the shape of the SWAP, but does not increase the speed. 

DONOVAN AND WILSON Solitary Wave Acoustic Polaron Motion... 

Earlier than all of this Holstein (10) described his optic polaron, in which the deformation of the ID chain is an optic deformation. His was the first polaron solution which was a Solitary Wave or Soliton. In such a deformation there is no change in lattice density in the polaron there is only a rearrangement of atoms without a change of density. In the case of the optic polaron there is no analytical solution for the moving polaron. However it is clear that on increase of the optic polaron energy due to motion there is no perturbation as the velocity goes through the sound velocity. In the pure optic polaron the sound velocity is not in the model at the outset. It is in the motion of the polaron at velocities up to the sound velocity that the profound difference between the acoustic and optic polarons occurs the difference in properties of the polarons at rest is leas Important. 

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... 

Finally, solitary waves are characterized by their collisions. There exist two main types of wave collisions, oblique and head-on collisions, which generate different patterns in the liquid surface. Head-on collisions are better analyzed in a space-time diagram, whereas oblique collisions can be easily analyzed in real space. 

Weidman, P.D., Linde, H., and Velarde, M.G., Evidence for solitary wave behavior in Marangoni-Benard convection, Phys. Fluids A, 4, 921-926, 1992. 

Nepomnyashchy, A. A. and Velarde, M.G., A three-dimensional description of solitary waves and their interaction in Marangoni-Benard layers, Phys. Fluids, 6, 187-198, 1994. 

Linde, H. et al.. Interfacial wave motions due to Marangoni instabihty. ni. Solitary waves and [periodic] wave trains and then collisions and reflections leading to dynamic network [cellular] patterns in large containers, J. Colloid Interface Sci., 236, 214—224, 2001. 

Experimental Evidence for Solitary-Wave Excitations in GCP (Abstract only) 625... 

M. Bar, M. Eiswirth, H. H. Rotermund, and G. Ertl. Solitary wave phenomena in an excitable surface reaction. Physical Review Letters, 69 945-948, 1992. 

Experiments will be necessary to prove the existence, behavior, and engineering of a new elass of physical instability. Tensor solitary waves have been hypothesized that are related to debonding instabilities first deteeted in particulate eomposites in the early 1980 s. Figure 4 shows the eharacteristies of that simpler instability. Figure 5 eaptures a mysterious, ultrafast failure mode first observed in 2000, whose explanation may be similar to Figure 4 s partieulate (not fiber) composite results, but whose... 

However, in vivo it can easily happen (by collisions of solute molecules of the cytoplasma or by enzymatic action) that the carcinogen becomes cut off. The question then arises whether the system (together with the changed water structure and ionic distribution) immediately relaxes, or a solitary wave starts to travel in both directions from the original disturbance (see Figure 6). 

Figure 6 After in vivo detachment of the carcinogen, the coupled geometrical distortion and change in the stacking interaction starts to travel along the chain in both directions (indicated by arrows) as a non-linear solitary wave...

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