SOLITONS

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SOLITONS From Canal Water Waves to Molecular
Lasers
Hieu D. Nguyen Rowan University
IEEE Night 5-20-03
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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
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Definition of Soliton
One entry found for soliton. Main Entry
soliton   Pronunciation '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
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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
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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
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(No Transcript)
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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.
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Copperplate etching by J. Scott Russell depicting
the 30-foot tank he built in his back garden in
1834
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Controversy Over Russells 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/Mathemat
icians/Russell_Scott.html
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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

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Korteweg-de Vries (KdV) Equation
Rescaling
KdV Equation
Nonlinear Term
Dispersion Term
(Steepen)
(Flatten)
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Stable Solutions
Profile of solution curve

• Unchanging in shape
• Bounded
• Localized

Do such solutions exist?
Steepen Flatten Stable
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Solitary Wave Solutions
1. Assume traveling wave of the form
2. KdV reduces to an integrable equation
3. Cnoidal waves (periodic)
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4. Solitary waves (one-solitons)
- Assume wavelength approaches infinity
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Other Soliton Equations
Sine-Gordon Equation

• Superconductors (Josephson tunneling effect)
• Relativistic field theories

Nonlinear Schroedinger (NLS) Equation

• Fiber optic transmission systems
• Lasers

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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
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Inverse Scattering
Nonlinear Fourier Transform
Space-time domain
Frequency domain
Fourier Series
http//mathworld.wolfram.com/FourierSeriesSquareWa
ve.html
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Solving Linear PDEs by Fourier Series
1. Heat equation
2. Separate variables
3. Determine modes
4. Solution
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Solving Nonlinear PDEs by Inverse Scattering
1. KdV equation
2. Linearize KdV
3. Determine spectrum
(discrete)
4. Solution by inverse scattering
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2. Linearize KdV
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Schroedingers Equation (time-independent)
Potential (t0)
Eigenvalue (mode)
Eigenfunction
Scattering Problem
Inverse Scattering Problem
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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
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4. Solution by Inverse Scattering
(a) Solve GLM integral equation (1955)
(b) N-Solitons (GGKM, WT, 1970)
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Soliton matrix
One-soliton (N1)
Two-solitons (N2)
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Unique Properties of Solitons
Signature phase-shift due to collision
Infinitely many conservation laws
(conservation of mass)
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Other Methods of Solution
Hirota bilinear method Backlund
transformations Wronskian technique Zakharov-Sha
bat dressing method
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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

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Applications of Solitons
Optical Communications
- Temporal solitons (optical pulses)
Lasers

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

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Hieu Nguyen Temporal solitons involve weak
nonlinearity whereas spatial solitons involve
strong nonlinearity
Optical Phenomena
Refraction
Diffraction
Coherent Light
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NLS Equation
Nonlinear term
Dispersion/diffraction term
One-solitons
Envelope
Oscillation
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Temporal Solitons (1980)
Chromatic dispersion
- Pulse broadening effect
Before
After
Self-phase modulation
- Pulse narrowing effect
Before
After
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Spatial Solitons
Diffraction
- Beam broadening effect

Self-focusing intensive refraction (Kerr effect)
- Beam narrowing effect
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BEC (1995)
Cold atoms

• Coherent matter waves
• Dilute alkali gases

http//cua.mit.edu/ketterle_group/
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Atom Lasers
Atom beam
Gross-Pitaevskii equation
- Quantum field theory
Atom-atom interaction
External potential
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Molecular Lasers
Cold molecules
- Bound states between two atoms (Feshbach
resonance)
Molecular laser equations
(atoms)
(molecules)
Joint work with Hong Y. Ling (Rowan University)
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Many Faces of Solitons
Quantum Field Theory

• Quantum solitons
• Monopoles
• Instantons

General Relativity

• Bartnik-McKinnon solitons (black holes)

Biochemistry
- Davydov solitons (protein energy transport)
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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
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Recreation of the Wave of Translation (1995)
Scott Russell Aqueduct on the Union Canal near
Heriot-Watt University, 12 July 1995
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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/s
olitons/solitons.html Alkali Gases _at_ Mit Home
page http//cua.mit.edu/ketterle_group/
www.rowan.edu/math/nguyen/soliton/