Solitary waves and solitonsThe correct term for a wave which is localized and retains its form over a long period of time is: solitary wave. However, a soliton is a solitary wave having the additional property that other solitons can pass through it without changing its shape. But, in the literature it is customary to refer to the solitary wave as a soliton, although this is strictly incorrect Tao-08.Figure 15: Figure 15: Evolution of a two-soliton solution of the KdV equation. Image illustrates the collision of two solitons that are both moving from left to right. The faster (taller) soliton overtakes the slower (shorter) soliton.Solitons are stable, nonlinear pulses which exhibit a fine balance between non-linearity and dispersion. They often result from real physical phenomena that can be described by PDEs that are completely integrable, i.e. they can be solved exactly. Such PDEs describe: shallow water waves, nonlinear optics, electrical network pulses, and many other applications that arise in mathematical physics. Where multiple solitons moving at different velocities occur within the same domain, collisions can take place with the unexpected phenomenon that, first they combine, then the faster soliton emerges to proceed on its way. Both solitons then continue to proceed in the the same direction and eventually reach a situation where their speeds and shapes are unchanged. Thus, we have a situation where a faster soliton can overtake a slower soliton. There are two effects that distinguishes this phenomena from that which occurs in a linear wave system. The first is that the maximum height of the combined solitons is not equal to the sum of the individual soliton heights. The second is that, following the collision, there is a phase shift between the two solitons, i.e. the linear trajectory of each soliton before and after the collision is seen to be shifted horizontally – see figure (15).Some additional discussion is given in section (The Korteweg-de Vries equation) and detailed technical overviews of the subject can be found in the works by Ablowitz & Clarkson Abl-91, Drazin & Johnson Dra-89 and Johnson Joh-97. Soliton theory is still an active area of research and a discussion on the various types of soliton solution that are known is given by Gerdjikov & Kaup Ger-05.Soliton typesSoliton types generally fall into thee types:Humps (pulses) – These are the classic bell-shaped curves that are typically associated with soliton phenomena.Kinks – These are solitons characterized by either a monotonic positive shift (kink) or a monotonic negative shift (anti-kink) where the change in value occurs gradually in the shape of an s-type curve.Breathers (bions) – These can be either stationary or travelling soliton humps that oscillate: becoming positive, negative, positive and so on.More details may be found in Drazin and Johnson Dra-89.TsunamiThe word tsunami is a Japanese term derived from the characters ? (tsu) meaning harbor and ? (nami) meaning wave. It is now generally accepted by the international scientific community to describe a series of traveling waves in water produced by the displacement of the sea floor associated with submarine earthquakes, volcanic eruptions, or landslides. They are also known as tidal waves.Tsunami are usually preceded by a leading-depression N-wave (LDN), one in which the trough reaches the shoreline first. Eyewitnesses in Banda Aceh who observed the effects of the December 2004 Sumatra Tsunami, see figure (9), resulting from a magnitude 9.3 seabed earthquake, described a series of three waves, beginning with a leading depression N wave Bor-05. Recent estimates indicate that this powerful tsunami resulted in excess of 275,000 deaths and extensive damage to property and infrastructure around the entire coast line of the Indian ocean Kun-07.Tsunami are long-wave phenomena and, because the wavelengths of tsunami in the ocean are long with respect to water depth, they can be considered shallow water waves. Thus, cp=cg=gh??? and for a depth of 4km we see that the wave velocity is around 200 m/s. Hence, tsunami waves are often modelled using the shallow water equations, the Boussinesq equation, or other suitable equations that bring out in sufficient detail the required wave characteristics. However, one of the major challenges is to model shoreline inundation realistically, i.e. the effect of the wave when it encounters the shore – also known as run-up. As the wave approaches the shoreline, the water depth decreases sharply resulting in a greatly increased surge of water at the point where the wave strikes land. This requires special modeling techniques to be used, such as robust Riemann solvers Tor-01,Ran-06 or the level-setmethod Set-99,Osh-03, which can handle situations where dry regions become flooded and vice versa.