Slow variables in Nonlinear Schrodinger Equation

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SUMMARY

The discussion centers on the concept of "slow variables" in the Nonlinear Schrödinger Equation (NLS), specifically as presented in R.S. John's book "A Modern Introduction to the Mathematical Theory of Water Waves." The slow variables are defined as zeta = epsilon * (x - ct) and T = epsilon * t, where epsilon is a small parameter. This framework allows for the approximation of coefficients in the equation as constant due to their slow variation, which is crucial for understanding wave packet modulation in hydrodynamics.

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  • Understanding of Nonlinear Schrödinger Equation (NLS)
  • Familiarity with hydrodynamics concepts
  • Knowledge of perturbation methods in mathematical physics
  • Basic grasp of wave packet theory
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Physicists, mathematicians, and students studying wave dynamics, particularly those interested in the applications of the Nonlinear Schrödinger Equation in hydrodynamics and quantum theory.

hanson
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Hi all.
What do it mean by "slow variables" in NLS?
I am reading a derivation of the NLS in the context of hydrodynamics, by R.S.John in his book "A modern introduction to the Mathematical Theory of Water Waves".

In the book, slow variables are zeta = epsilon * (x-ct) and T = epsilon * t.
It is counter-intuitive to me...
It seems that when epsilon is very small, the reference frame in the new time T shall observes a much faster motion then in the frame of t, right?

Why is it called "slow variables"?

What is "slow modulation of the wave packets" by the way?
 
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This isn't really a mathematics question. I'm moving it to "Physics- Quantum Theory" where it may get more replies.

What it means, of course, is coefficients in the equation that are changing slowly enough that, in the first approximation, they can be taken to be constant.
 
changing coefficients? I still don't quite get it...
 

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