Soliton Theory (part 2 of 2)

Besides the KdV equation covered last time, some well known soliton equations are:

-Nonlinear Schrodinger equation (NLSE):

i ∂_t ψ = - ½ ∂_x² ψ + k |ψ|² ψ

-Sine Gordon equation:

            φ_tt – φ_xx + sin φ = 0

Kadomtsev-Petiashvili equation:

            ∂_x (∂_t u + u ∂_x u + 2ε² ∂_xxx u) ± ∂_yy u = 0

I am most familiar with the nonlinear Schrodinger equation and its variations. NLSE was proposed some 40 years ago to be used for describing light propagation in optical fibers. People used signal fires

to transmit information since ancient times (or since Middle Earth J ) but one needs guided light transmission to eliminate interference from the atmosphere (rain, fog, etc). However using ordinary glass is not practical because ordinary glass is not transparent enough. Imagine looking through a glass 30 miles thick!!! 30 miles is the typical length for an optical fiber because it takes dissipation 30 miles to reduce the intensity in half and at that point signal amplification is required. Advances in glass manufacturing resulted in ultra-transparent optical fibers (at certain wavelengths) close to the theoretical transparency limit (this limit is due to Reyleigh scattering which is responsible for the color blue of the sky).

From Maxwell’s equations, one can derive NLSE. The anharmonic electron oscillation generates the nonlinear |ψ|² ψ term. The ∂_x² ψ term corresponds to ordinary dispersion in the optical fiber and a soliton is a “light bullet” which carries one bit. Using solitons, a single optical fiber can carry giga (10⁹) bits of information per second (the usual soliton pulse is measured in pico-seconds (10^-9 s) but femto-second pulses for shorter distances are possible too). A single telephone call requires 64 kbit/s and a 1GB/s optical line can carry 15,625 concurrent calls. In 1999 rates of 300GB/s for a single fiber were commercially achieved and an optical cable bundle has much more than a single optical fiber. Advances in long distance transmission rates were so great that despite Moore’s law computers started to be viewed as the hopeless communication bottleneck.

For all their potential, optical solitons never materialize in practice and there is this funny disconnect between academic research and industry which I encountered several times. For example after I graduated I had a job interview at Bell Labs hoping to amaze them with my NLSE research, but using low intensity traditional pulses, they already achieved in practice one order of magnitude higher transmission rate than what was demonstrated with solitons at that time (based on publications in experimental journals). Solitons are not very practical due to arriving time instabilities and the need to use expensive active pulse repeaters every 30 miles instead of passive amplification (imagine an active repeater malfunctioning at the bottom of Pacific Ocean in an undersea cable).

Another academia-industry disconnect I encountered was the one on training a neural network for OCR (optical character recognition). Just like soliton theory generated thousands of research papers but the industry never adopted it, in neural networks there is an extensively studied “back propagation method”. In practice this is all useless and the industry has much more effective techniques for training neural networks, but they are all trade secrets. With the industry knowledge, reading the back propagation papers made for a good laugh.

But solitons are nice topic in themselves and they do have very interesting math properties.

In practice there are two standard techniques for computing soliton solutions: the Lax pair and the zero curvature condition. The Lax pair is linked with a Sturm-Liouville equation, and solitonic equations have an infinite number of conservation laws. Discovering a lot of conserved quantities for a new equation is a sure clue that the equation admits soliton solutions.

The KdV equation can be expressed as a Hamiltonian system in two distinct ways. In fact such a case is called bi-Hamiltonian, and the interplay between the two Hamitonians generates an infinite number of conservation laws.

Another way one can solve the solitonic equations is by the Riemann-Hilbert problem. Through this problem there is an unexpected link between solitons and renormalization in field theory.

In an initial value problem for solitonic equations, part of the initial condition excites dispersive waves, and (if the energy is large enough) part excites the solitonic pulses. To solve the Riemann-Hilbert problem, given a closed curve on the Riemann sphere (the complex plane + infinity) one has to marry two functions on each side of the curve given some initial value on the curve. In soliton land, the initial value on the curve corresponds to the dispersive waves, and solitons correspond to poles on the Riemann sphere. This is why solitons are robust: once there, the poles cannot be eliminated (in the absence of friction or additional effects which destroys the infinite number of conservation laws).

An excellent review on the Riemann-Hilbert problem and solitons can be found here. The gist of the Riemann-Hilbert problem is: reconstruct an analytic function from its singularities. As Alexander Its points out, in its most general way, integrability means that local properties (singularities) determine global behavior.

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