Saturday, December 7, 2013

Soliton theory (part 1 of 2)

In the last post I listed the amazing lectures of Mr. Bender on perturbation theory. If you managed to follow the lectures to the end, you got to see the WKB perturbation theory in action. The lecture ending was on extracting a “beyond all orders” behavior.

After watching a powerful movie, don’t you want sometimes to have a different ending? For example the movie Inception:

If you have not watch it, I won’t spoil it by saying more, but if you did, you understand what I mean.

So I cannot help and I’ll attempt to give an alternative ending to Mr. Bender lectures, by venturing into the wonderful area of soliton theory.

In lessons 13, 14 and 15 you got to see how to solve a potential well in Schrodinger’s equation using WKB. At each turning point there is a reflected wave, and one may ask: are there potential well for which there is no reflection? How would a reflectionless potential look like? This is the starting point of the so-called Inverse Scattering Theory .

But let’s start in historical fashion.

In 1832 a certain gentleman, Mr. John Scott Russell, got amazed by a peculiar wave along an English canal:

I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of great agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel.”

It was not until 1895 when Diederik Korteweg and Gustav deVries derived the equation of this wave:

U_t + 6 U U_x + U_xxx = 0

(now called the KdV equation)

Because the equation contains the second power of U it is a nonlinear equation, and in particular it is a nonlinear partial differential equation, an ugly beast of intractable complexity which nobody knew how to solve.

Fast forward to 1952: Fermi, Pasta, and Ulam were doing computer modeling for a certain problem and noticed some odd periodic behavior.

Further investigating this in 1965, Zabusky and Kruskal observed the kind of solitary waves Mr. Russell witnessed. Those solitary waves or pulses were able to pass through one another with no perturbation whatsoever (and hence the term solitons) which was a very bizarre behavior.

 The mathematical breakthrough occurred in 1967 when Gardner, Green, Kruskal and Miura discovered the inverse scattering technique for solving the KdV equation.

But what is inverse scattering? Let us start simpler with linear partial differential equations. How do we solve the initial value problem? The standard technique is that of a Fourier transform.

In a Fourier transform we multiply a function of x: f(x) with exp(ikx) and then we integrate over all x. What results if a function F(k). Fine but what does this have to do with partial differential equations?

Let us take the Fourier transform on the linear partial differential equation. Wherever we have a partial derivation, we integrate by parts and transfer the derivation to the exp(ikx) term. In turn this extracts the iK factor out of the exponent, and under the integral sign we transformed the linear partial differential equation into a polynomial equation in K. Wow (applause please)!!!

Solving polynomial equations is MUCH easier than solving partial differential equations.

So the general technique is the following: extract the (Fourier) modes, solve the easy time problem in (Fourier) modes, and perform a reversed (Fourier) transform to obtain the solution at a later time:

K(t=0)à-solve time evolution in polynomial equation-à---K(t=T)
  ^                                                                                                 \/
  ^                                                                                                 \/
  ^                                                                                                 \/
F(K(t=0))                                                                               F(K(t=T))
  ^                                                                                                 \/
  ^                                                                                                 \/
Fourier Transform                                                       Inverse Fourier Transform
  ^                                                                                                 \/
  ^                                                                                                 \/
f(x(t=0))                                                                                    f(x(t=T))

Something similar happens in nonlinear partial differential equation and the role of the Fourier transform is taken by solving a Schrodinger equation scattering problem. The scattering potential in the Schrodinger equation is the solution to the nonlinear partial differential equation. The typical solitonic solution is of the form 1/cosh. Solving nonlinear partial differential equations takes this form:

 S(t=0)à-solve time evolution in polynomial equation-à---S(t=T)
  ^                                                                                                 \/
  ^                                                                                                 \/
  ^                                                                                                 \/
scattering parameters(t=0)                                      scattering parameters(t=T)
  ^                                                                                                 \/
  ^                                                                                                 \/
Direct Scattering Problem                                        Inverse Scattering Problem
  ^                                                                                                 \/
  ^                                                                                                 \/
 f(x(t=0))                                                                                    f(x(t=T))

Solving the direct scattering problem is straightforward, and here one may use WKB for example, but in practice this does not happen. Using WKB to derive the reflectionless property (this is why solitons pass through each other unperturbed) would yield the 1/cosh solution.

The tricky problem is the inverse scattering by solving a Gelfand-Levitan-Marchenko problem. This is where the computation becomes intensive. The technique makes use of Jost functions to define the boundary conditions, but those are details.

Solitons theory is a very nice and rich area with unexpected links in mathematics and physics. Next time I’ll present some famous solitonic equations, the unexpected link with renormalization theory, and the real world potential usefulness of solitons.

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