Lecture 20, 18.376, Tue Apr 25, 2023.
Summary
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Next few lectures will be based on the Fourier Transform solution to
dispersive wave systems. Advantage of the Fourier Transform is that it
provides a very detailed and generic access to the properties of
dispersive wave systems. Disadvantage, relative to perturbative methods
like multiple-scales, is that it only works for linear systems ---
multiple scales (or average Lagrangian) works for nonlinear problems,
or at least, some nonlinear problems.

While this course is mostly about the theory for linear dispersive waves,
at this point it is worth asking how much of the theory can be extended
to nonlinear systems? A rough answer is:
1) For weakly nonlinear systems, quite a bit (though not everything).
   It is very hard to generalize things that depend too intimately on
   details of the Fourier Transform. For example, subjects like turning
   points, caustics, and others that have to do with transitions from
   waves to no waves are extremely difficult to deal with once even a
   moderate amount of nonlinearity is introduced. A typical problem that
   arises is that this situations trigger relatively large amplitudes and
   large derivatives, which often are "just right" to make the
   nonlinearity kick in (even if weak in principle). For a simple example
   of this, see
      Caustics of weak shock waves.
      Physics of Fluids, 10:206-222, (1998).
   This is not for a dispersive system [dispersive systems are way
   harder, often intractable], but it illustrates the issue. For examples
   that illustrate a bit what can happen with dispersive systems, check
   the literature on "The small dispersion limit", which explores the
   question of what happens when you add a small amount of dispersion to
   a nonlinear system.
2) For fully nonlinear systems not much in general. The only "general"
   thing there is the work of Whitham on modulation of a single frequency
   traveling wave.
   For "special" systems, solvable via the "Inverse Scattering Transform",
   there is a fair amount ... for this systems, effectively, a "nonlinear
   Fourier Transform" exists. But these systems are very, very, special!

So, why are nonlinear systems so hard to deal with? The core issue is the
so called "small divisors problem" ... this was explained in the lecture
and I will not summarize it here. It is a very old problem which is still
full of unsolved questions. It is "the source" of chaos in dispersive type
systems, and the main theory that deals with it is KAM theory. The special systems named in (2) are special because they manage to completely avoid
this problem, which for "arbitrary" systems happen with "probability zero"
(take this with a grain of salt, though).

OK, back to the subject matter: % ==========================================
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Cover #021b "Fourier Transform Approach [narrow band spectrum]."

Start review of the Stationary Phase method
Cover the "Stationary phase and the far field approximation"
Section 3 in the "Lecture Topics" notes posted on the web page.
   Covered first part of "3.1 Large time for a 1-D scalar equation"
   up to formula (3.2).
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