Hi all. Can someone explain me physically why we need to deal with wave packets in water waves? I know the the nonlinear schrodinger equations deals with wave packets in water wave. But why bother dealing with wave packets? For the KdV equation, the concept of wave packets is not needed, why? What so special about wave packets? Please help.
Recommend a good book The theory of solitons is a beautiful, intricate, and highly developed subject, so anyone who wants to know more should consult a good book since there is a lot to learn if you want to understand the basics. However, IMO it is not nearly as confusing as hanson makes out! I like the undergraduate level introduction by P. G. Drazin and R. S. Johnson, Solitons: an Introduction, Cambridge University Press, 1989. If you follow this up, you will see how the usual Schroedinger equation plays a role in the famous inverse-scattering transform method of solving the KdV, a famous soliton equation, which arises as an approximation of water waves under certain circumstances. This spawned a great deal of work, including analysis of related PDEs, such as the mKdV, the BBM equation, the Camassa-Holm equation, etc. (the latter also arises as approximations of water waves and includes idealized "breaking of waves"; see math.AP/0709.0905). The nonlinear Schroedinger equation is a nonlinear generalization of the Schroedinger equation which itself has some aspects of a soliton equation. The sine-Gordon equation is another well known nonlinear PDE which has some soliton-like solutions. You can search the arXiv to find many recent papers discussing current research in this area. Needless to say, you will need a strong background in differential equations to follow this research.