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from SIAM News, Volume 31, Number 2, 1998

Making Waves: Solitons and

Their Practical Applications

"A Bright Idea“Economist (11/27/99) Vol. 353, No. 8147, P. 84

Solitons, waves that move at a constant shape and speed, can

be used for fiber-optic-based data transmissions…

From the Academy

Mathematical frontiers in optical solitons

Proceedings NAS, November 6, 2001

Number 588, May 9, 2002 Bright Solitons in a Bose-Einstein Condensate

Solitons may be the wave of the future Scientists in

two labs coax very cold atoms to move in trains

05/20/2002

The Dallas Morning News

Definition of ‘Soliton’

One entry found for soliton.

Main Entry: sol·i·tonPronunciation: 'sä-l&-"tänFunction: nounEtymology: solitary + 2-onDate: 1965: a solitary wave (as in a gaseous plasma) that propagates

with little loss of energy and retains its shape and speed

after colliding with another such wave

http://www.m-w.com/cgi-bin/dictionary

Solitary Waves

John Scott Russell (1808-1882)

- Scottish engineer at Edinburgh
- Committee on Waves: BAAC

Union Canal at Hermiston, Scotland

http://www.ma.hw.ac.uk/~chris/scott_russell.html

Great Wave of Translation

“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 violent 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…”

- J. Scott Russell

“…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. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.”

“Report on Waves” - Report of the fourteenth meeting of the British Association

for the Advancement of Science, York, September 1844 (London 1845), pp 311-390,

Plates XLVII-LVII.

Copperplate etching by J. Scott Russell depicting the 30-foot tank he built in his back garden in 1834

Controversy Over Russell’s Work1

George Airy:

- Unconvinced of the Great Wave of Translation
- Consequence of linear wave theory

G. G. Stokes:

- Doubted that the solitary wave could propagate

without change in form

Boussinesq (1871) and Rayleigh (1876);

- Gave a correct nonlinear approximation theory

1http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Russell_Scott.html

Model of Long Shallow Water Waves

D.J. Korteweg and G. de Vries (1895)

- surface elevation above equilibrium
- depth of water
- surface tension
- density of water
- force due to gravity
- small arbitrary constant

Korteweg-de Vries (KdV) Equation

Rescaling:

KdV Equation:

Nonlinear Term

Dispersion Term

(Steepen)

(Flatten)

Stable Solutions

Profile of solution curve:

- Unchanging in shape
- Bounded
- Localized

Do such solutions exist?

Steepen + Flatten = Stable

Solitary Wave Solutions

1. Assume traveling wave of the form:

2. KdV reduces to an integrable equation:

3. Cnoidal waves (periodic):

4. Solitary waves (one-solitons):

- Assume wavelength approaches infinity

Other Soliton Equations

Sine-Gordon Equation:

- Superconductors (Josephson tunneling effect)
- Relativistic field theories

Nonlinear Schroedinger (NLS) Equation:

- Fiber optic transmission systems
- Lasers

N-Solitons

Zabusky and Kruskal (1965):

- Partitions of energy modes in crystal lattices
- Solitary waves pass through each other
- Coined the term ‘soliton’ (particle-like behavior)

Two-soliton collision:

Inverse Scattering

“Nonlinear” Fourier Transform:

Space-time domain

Frequency domain

Fourier Series:

http://mathworld.wolfram.com/FourierSeriesSquareWave.html

Solving Linear PDEs by Fourier Series

1. Heat equation:

2. Separate variables:

3. Determine modes:

4. Solution:

Solving Nonlinear PDEs by Inverse Scattering

1. KdV equation:

2. Linearize KdV:

3. Determine spectrum:

(discrete)

4. Solution by inverse scattering:

Schroedinger’s Equation

(time-independent)

Potential

(t=0)

Eigenvalue

(mode)

Eigenfunction

Scattering Problem:

Inverse Scattering Problem:

3. Determine Spectrum

(a) Solve the scattering problem at t = 0 to obtain

reflection-less spectrum:

(eigenvalues)

(eigenfunctions)

(normalizing constants)

(b) Use the fact that the KdV equation is isospectral

to obtain spectrum for all t

- Lax pair {L, A}:

4. Solution by Inverse Scattering

(a) Solve GLM integral equation (1955):

(b) N-Solitons ([GGKM], [WT], 1970):

Unique Properties of Solitons

Signature phase-shift due to collision

Infinitely many conservation laws

(conservation of mass)

Other Methods of Solution

Hirota bilinear method

Backlund transformations

Wronskian technique

Zakharov-Shabat dressing method

Decay of Solitons

Solitons as particles:

- Do solitons pass through or bounce off each other?

Linear collision:

Nonlinear collision:

- Each particle decays upon collision
- Exchange of particle identities
- Creation of ghost particle pair

Applications of Solitons

Optical Communications:

- Temporal solitons (optical pulses)

Lasers:

- Spatial solitons (coherent beams of light)
- BEC solitons (coherent beams of atoms)

Hieu Nguyen:

Temporal solitons involve weak nonlinearity whereas spatial solitons involve strong nonlinearity

Optical Phenomena

Refraction

Diffraction

Coherent Light

Temporal Solitons (1980)

Chromatic dispersion:

- Pulse broadening effect

Before

After

Self-phase modulation

- Pulse narrowing effect

Before

After

Spatial Solitons

Diffraction

- Beam broadening effect:

Self-focusing intensive refraction (Kerr effect)

- Beam narrowing effect

Atom Lasers

Atom beam:

Gross-Pitaevskii equation:

- Quantum field theory

Atom-atom interaction

External potential

Molecular Lasers

Cold molecules

- Bound states between two atoms (Feshbach resonance)

Molecular laser equations:

(atoms)

(molecules)

Joint work with Hong Y. Ling (Rowan University)

Many Faces of Solitons

Quantum Field Theory

- Quantum solitons
- Monopoles
- Instantons

General Relativity

- Bartnik-McKinnon solitons (black holes)

Biochemistry

- Davydov solitons (protein energy transport)

Future of Solitons

"Anywhere you find waves you find solitons."

-Randall Hulet, Rice University, on creating solitons in

Bose-Einstein condensates, Dallas Morning News, May 20, 2002

Recreation of the Wave of Translation (1995)

Scott Russell Aqueduct on the Union Canal

near Heriot-Watt University, 12 July 1995

References

C. Gardner, J. Greene, M. Kruskal, R. Miura, Korteweg-de Vries equation and generalizations. VI.

Methods for exact solution, Comm. Pure and Appl. Math. 27 (1974), pp. 97-133

R. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Review 18 (1976), No. 3, 412-459.

A. Snyder and F.Ladouceur, Light Guiding Light, Optics and Photonics News, February, 1999, p. 35

P. D. Drummond, K. V. Kheruntsyan and H. He, Coherent Molecular Solitons in Bose-Einstein

Condensates, Physical Review Letters 81 (1998), No. 15, 3055-3058

B. Seaman and H. Y. Ling, Feshbach Resonance and Coherent Molecular Beam Generation in a Matter

Waveguide, preprint (2003).

H. D. Nguyen, Decay of KdV Solitons, SIAM J. Applied Math. 63 (2003), No. 3, 874-888.

M. Wadati and M. Toda, The exact N-soliton solution of the Korteweg-de Vries

equation, J. Phys. Soc. Japan 32 (1972), no. 5, 1403-1411.

Solitons Home Page: http://www.ma.hw.ac.uk/solitons/

Light Bullet Home Page: http://people.deas.harvard.edu/~jones/solitons/solitons.html

Alkali Gases @ Mit Home page: http://cua.mit.edu/ketterle_group/

www.rowan.edu/math/nguyen/soliton/

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