Slow variables in Nonlinear Schrodinger Equation

In summary, the conversation discusses the concept of "slow variables" in the context of NLS and hydrodynamics. These variables, zeta and T, are defined as epsilon times (x-ct) and epsilon times t, respectively. It is noted that this may seem counter-intuitive as the reference frame in T would observe a faster motion than in t. Additionally, there is a question about the term "slow modulation of the wave packets" and its meaning. The conversation is then redirected to a different topic and moved to a different forum. The concept of slow variables refers to coefficients in the equation that change slowly enough to be considered constant in the first approximation.
  • #1
hanson
319
0
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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  • #2
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.
 
  • #3
changing coefficients? I still don't quite get it...
 

1. What is the Nonlinear Schrodinger Equation?

The Nonlinear Schrodinger Equation (NLSE) is a partial differential equation that describes the evolution of a wave packet in a nonlinear medium. It is used in various fields of physics, such as optics, plasma physics, and condensed matter physics, to study the behavior of complex systems.

2. What are slow variables in the Nonlinear Schrodinger Equation?

Slow variables in the Nonlinear Schrodinger Equation refer to the parameters that change slowly compared to the characteristic time scale of the system. In this case, the NLSE is used to study the long-term behavior of the system, while the slow variables are considered as external parameters.

3. How are slow variables incorporated into the Nonlinear Schrodinger Equation?

To incorporate slow variables into the Nonlinear Schrodinger Equation, one can use the adiabatic approximation, where the slow variables are treated as constants over a short time interval. This simplifies the equation and allows for easier analysis of the system's behavior.

4. What is the significance of studying slow variables in the Nonlinear Schrodinger Equation?

Studying slow variables in the Nonlinear Schrodinger Equation allows for a deeper understanding of the dynamics of complex systems. It can also provide insights into how external parameters can affect the behavior of the system over time, and how these changes can be controlled or manipulated.

5. Are there any real-world applications of the Nonlinear Schrodinger Equation and slow variables?

Yes, the Nonlinear Schrodinger Equation and the concept of slow variables have various real-world applications in fields such as optics, where they are used to study the behavior of light in nonlinear media. They are also used in the study of Bose-Einstein condensates, superfluids, and other physical systems.

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