Soliton-like solution

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... 

Carlson et al. 1994, Schmittbhul et al 1993, Vilotte et al 1994). In fact, Schmitbhul et al (1993) noted that the system shows earthquake-like almost chaotic solutions for very small values of = Nv/y/R while the solutions are soliton-like for 0 of the order of unity. This size-dependent transition has been extensively studied by Vilotte et al (1994) and Anan-thakrishna and Ramachandran (1994), who also showed that the same transition occurs as the parameter a in (4.4) decreases from higher values. As the blocks in the model represent the independent junctions where the earth s crust rests on a moving tectonic plate (for any particular epicentre region, in case of earthquakes), the number N of such blocks in the model is usually quite small. Also, the (dimensionless) tectonic plate velocity v is very small in reality and of the order of 10 (Carlson et al 1994). This... 

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. 

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. 

In Figure 2 we show a space time plot of a CP mode for = 1. We note that there are collisions at ip = 0 and ip = it. The collision sites are determined by the initial conditions. This CP mode is periodic and symmetric in that the behavior at the two collision sites is the same, just 180° out of phase in time. In this mode, the hot spots enter and leave the collision essentially unchanged except for a phase shift, much like the behavior of solitons. Our computations indicate that the amplitude of the symmetric CP mode, i.e., the maximum temperature achieved at a collision, does not approach 0 as i approaches the transition point. Thus, the stable CP modes develop with finite amplitude. Our results also indicate that the mean speed V for these modes exceeds the speed for the unstable, uiuformly propagating planar solution. Thus, near the transition point finite amplitude CP modes can propagate faster than the uiuformly propagating mode. The mean propagation speed decreases as R increases. 

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