Why Do We Need Wave Packets in Water Waves?

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Wave packets are essential in the study of water waves because they provide analytical solutions to the nonlinear Schrödinger equation (NLSE), which is applicable in this context. Unlike the Korteweg-de Vries (KdV) equation, which does not require wave packets, the NLSE captures the dynamics of wave interactions and solitons more effectively. The theory of solitons, including the inverse-scattering transform method, is crucial for understanding these phenomena and their relation to water waves. Recommended literature, such as "Solitons: an Introduction" by Drazin and Johnson, can help deepen understanding of these concepts. Overall, wave packets play a significant role in modeling and analyzing complex water wave behavior.
hanson
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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.
 
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Schroedinger equation deals with water? That's new to me.
 
genneth said:
Schroedinger equation deals with water? That's new to me.

yes, nonlinear schrodinger equation can be used in water waves context.
Please help.
 
They're special in that they represent particles?
 
hanson said:
What so special about wave packets?
They are one of the very few analytical solutions to the NLSE.
 
Recommend a good book

genneth said:
Schroedinger equation deals with water? That's new to me.

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.
 
How does KdV equation deal with wave packets?
 

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