Solitons
Waves, man. They’re everywhere.
Introduction
The goal of this article is to give the reader knowledge of the existence, properties, mechanics of solitons and a technique to obtain them from a special class of nonlinear partial differential equations (PDEs) known as Integrable Systems. To motivate the existence of solitons and their properties, a brief, selected history of their discovery in disparate parts of the world at different times is discussed. Then I try to discuss the idea of decomposition of an intial condition using a transform in linear and nonlinear equations. I then describe Hamiltonian structure of the integrable systems and its space of configurations. Lastly I describe an example of a soliton gas describing integrable turbulence.
History: John Scott Russel
As the story goes, one day (early 1800s), Scottish naval engineer, John Scott Russel, was watching two horses pulling a boat into a dock in the river Thames when the boat hits the edge of a dock and sends a wave propagating outward. Russel observed that the wave was not dissipating — not losing speed or height. So he hops on his horse and rides along the river to watch how long this wave would travel for. He chased the wave for 8 or 9 miles before he lost sight of it in the water channels. He was the first to observe, document and begin the study of… solitons. Although, the name soliton came much later. Our first definition of soliton: a local-in-space and stable-in-time perturbative profile arising in shallow water channels, and describing light packets through any media. The name came from the fact that it is a solitary wave with particle-like properties (electron, proton, gluon, boson, etc). One such property is the presence of conservation of elastic energy in soliton collisions. Hence “soli-ton”.
History: FPUT experiments
Fast forward to mid 1900s in Los Alamos, New Mexico when Fermi, Pasta, Ulam, and Tsingou are theorizing about interactions between subatomic particles. They create their famous FPUT experiment: a 1D lattice of finitely many beads of fixed masses, where each bead is connected to its nearest neighbors by a spring with the same fixed spring constant. If you were to displace one of the beads from its equilibrium, the tension in the springs would also displace its neighbors. This is also a simple model for a guitar string. If the forces in the spring are governed by Hooke’s law, the dynamics of the released bead from tension is described by time evolution of a trigonometric decomposition (using Fourier Series). If one were to take the limit as the number of beads goes to infinity, the lattice becoming a continuum, one obtains the wave equation. Wave equation describes propagation of high frequency laser pulses, light packets, electro-magnetic flow (Maxwell’s equations), and even water waves.
FPUT vs Wave Equation
The subtle difference between the wave equation dynamics and the FPUT experiment (besides the number of beads) is in the modification of Hooke’s law. The four scientists needed to account for quantum effects and added a first nonlinear approximation to Hooke’s law (originally a linear equation, the modification is now quadratic). When the continuum limit is taken (number of beads goes to infinity in the finite interval), one obtains the Kortweg-De Vries (KDV) equation, one of the simplest nonlinear partial differential equations. This subtle difference gives rise to seemingly chaotic dynamics and branches in to the fascinating fields of integrable and ergodic systems. The scientists were interested in thermalization of the system — how long it takes for the system to reach a thermal equilibrium? Their numerical experiments exhibited quasi-periodicity: from some initial state, the system decomposes into seemingly chaotic dynamics and, after really long times, the system can become arbitrarily close to the original state, but might never return to it.
Intuitive Theory: Solving Wave Equations
In linear partial differential equations, a common technique to solve the wave equation is done in three steps:
- Decompose the initial condition into component parts,
- Evolve the component parts in time (a linear problem to solve).
- Recompose the components to obtain the evolved signal.
Decompositions and recompositions can be done using Transforms (Fourier, Laplace, Melin, etc). Transforms are unitary operators on Vector spaces. i.e. unitary operators only rotate or reflect vectors.
Component parts can be represented by Orthogonal Polynomials (Bessel, Legendre, Hermite, Chebyshev, etc), Trigonometric functions (Fourier series decomposition), or even some multiplicative combination of them. This is a rich area of study, since the choice of component parts depends on the geometry of the system described. The true solution of the wave equation is represented by a weighted combination (sum or integral) of the natural coordinate components. The weights can be thought of as an alternative representation (and the natural coordinates) of the solution (in a Vector space). This is can also be thought of as the superposition principle.
Integrable Systems
These are a class of differential equations making up a set of measure 0 in the set of all differential equations. Yet, they represent physically significant phenomena and for which real-world systems are modeled as perturbations from these systems.
The Self Dual Yang Mills (SDYM) is conjectured to be the mother of all integrable systems.
We can obtain important equations from reductions of the SDYM. For example, we can obtain
- the Nonlinear Schrodinger (NLS) equation — models quantum packets of pressure and light propagation through nonlinear media,
- the Sine-Gordon (SG) equation — models action potentials traveling along axons in neurons of the brain,
- or the Kortweg-De Vries (KDV) equation — models small-amplitude, large-width water waves in shallow pools.
A new definition of soliton: a Galilean invariant solution of an integrable system. Meaning, translation of a solution in time or space, does not change the shape of the solution.
Hamiltonian Systems
The term integrability comes from the use of integration for solving linear differential equations. An integrable equation generally contains nonlinear terms necessitating a more appropriate definition of integration. The appropriate definition of integrability is encompassed in the Liouville-Arnold theorem, which states that a Hamiltonian system, with N degrees of freedom possessing N constants of motion that mutually commute with respect to the Poisson bracket, has a phase space that can be represented by N action-angle coordinate pairs. Simply, the phase space, representing all possible configurations of a physical system, can be described by a high-dimensional torus with N holes. A unique configuration of the system is described using N radial coordinates and N angle coordinates, where the angle coordinates evolve linearly in time. We are effectively describing the evolution of a physical system by drawing a trajectory on a configuration (phase) space.
Integrable systems can be interpreted as infinite dimensional Hamiltonian systems.
Gelfand-Levitan-Marchenko (GLM) Equation
This equation is a mathematical result that is interpreted as a generalization of Fourier Transform for nonlinear PDEs. This equation is a transform and arises in the last step the Inverse Scattering Transform (IST) — the famous method developed for solving KDV by relating it to the NLS. The IST consists of 3 steps that are each linear problems
- The initial condition in spatial domain is mapped (or transformed) into a spectral function in the spectral domain.
- Spectral data evolves linearly in time.
- Transform the spectral function back to obtain an evolved initial condition in the spatial domain using the GLM
The GLM was designed to invert solutions of PDEs who’s temporal evolution commutes with the Schrodinger operator. This is a consequence of the structure of the Lax Pair.
Example: Black holes as particles in a box
Imagine the expansive universe as being a large room containing many blackholes (like billions of billions). We treat each black hole like a particle of a gas in a box. With this perspective, it is convenient to create a statistical description of the evolution. The evolution and description of black holes is given by Einstein field equations which can be shown to be integrable when the cosmological constant is zero. Then black holes are classified as singular solitons. When taking a planar cross-section of the density profile of a black hole, it blows up — a singularity. And I conjecture that these massive waves move at constant velocity in space (even if we don’t notice it). Their interactions are always pair-wise (a property of integrable systems). A beautiful simulation can be found here. Because of the deterministic structure of integrable systems (we can determine all states for all time), studying these solutions can answer questions about the origins of our universe.
Conclusion
In this article, I’ve described a history of soliton discovery by naval engineer John Scott Russel in 1800s. Then I described how an integrable system (the KDV) is discovered in the FPUT experiments in 1950s as a continuum limit of lattice equations. Next, I described solutions of linear wave equations in terms of a weighted combinations of natural decompositions of the system and use this to describe the power of a transform. Then I describe a generalization of the Fourier Transform, known as the Inverse Scattering Transform, for obtaining solutions to integrable systems. This transform depends on the GLM equation in its last step. Since Integrable systems are PDEs that are interpreted as Hamiltonian systems, solutions can be determined for all time. Lastly, I discussed a particular example of billions of black holes treated as a singular soliton gas for which statistical mechanics can be applied and for which origins of our universe could be explored.
About Me
I am a PhD candidate at the University of Arizona Department of Applied Mathematics where my field of expertise is integrable systems and soliton theory. I think the vast set of techniques needed to understand this theory to be beautiful and far reaching. This type of picture should be shared and more accessible to others. I try to be a representative of this freely accessible information. If others are interested in other aspects of nonlinear physics, I can share my knowledge on the mathematics describing it. I look forward to questions, comments and requests!
