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SOLITONSFrom Canal Water Waves to Molecular Lasers
Hieu D. NguyenRowan University
IEEE Night5-20-03
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…
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
From the Academy
Mathematical frontiers in optical solitonsProceedings NAS, November 6, 2001
One entry found for soliton.
Main Entry: sol·i·ton 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
Definition of ‘Soliton’
John Scott Russell (1808-1882)
Union Canal at Hermiston, Scotland
Solitary Waves
http://www.ma.hw.ac.uk/~chris/scott_russell.html
- Scottish engineer at Edinburgh- Committee on Waves: BAAC
“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
Great Wave of Translation
“…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 Associationfor 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:
1http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Russell_Scott.html
- 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
Model of Long Shallow Water Waves
D.J. Korteweg and G. de Vries (1895)
22
2
3 1 2 1
2 2 3 3
g
t l x x
- surface elevation above equilibrium- depth of water- surface tension- density of water- force due to gravity- small arbitrary constant
lTg
31
3
Tll
g
6 0t x xxxu uu u
Nonlinear Term Dispersion Term6 0t xu uu 0t xxxu u
Korteweg-de Vries (KdV) Equation
3 2, , 2
2 3
g xt t x u
l
Rescaling:
KdV Equation:
(Steepen) (Flatten)
t
x
uu
tu
ux
Stable Solutions
Steepen + Flatten = Stable
- Unchanging in shape- Bounded- Localized
Profile of solution curve:
Do such solutions exist?
Solitary Wave Solutions
1. Assume traveling wave of the form:
( , ) ( ),u x t U z z x ct
3
36 0
dU dU d Uc U
dz dz dz
2. KdV reduces to an integrable equation:
3. Cnoidal waves (periodic):
2( ) cn ,U z a bz k
2 2 2 2( , ) 2 sech ( 4 ) ) , 4u x t k k x k t c k
4. Solitary waves (one-solitons):
2( ) sech2 2
c cU z z
- Assume wavelength approaches infinity
x
- u
x
- u
x
- u
x
- u
x
- u
Other Soliton Equations
Sine-Gordon Equation:
sinxx ttu u u
- Superconductors (Josephson tunneling effect) - Relativistic field theories
Nonlinear Schroedinger (NLS) Equation:
20t xxiu u u u
- Fiber optic transmission systems- Lasers
N-Solitons
- Partitions of energy modes in crystal lattices- Solitary waves pass through each other- Coined the term ‘soliton’ (particle-like behavior)
Zabusky and Kruskal (1965):
Two-soliton collision:
Inverse Scattering
“Nonlinear” Fourier Transform:
Space-time domain Frequency domain
Fourier Series:
01
( ) cos sinn nn
n x n xf x a a b
L L
http://mathworld.wolfram.com/FourierSeriesSquareWave.html
4 1 1( ) sin sin 3 sin 5 ...
3 5f x x x x
2 (0, ) ( , ) 0,
( ,0) ( )t xx
u t u L tu c u
u x f x
1. Heat equation:
2
1
0
( , ) sin
2( )sin
cnt
Ln
n
L
n
n xu x t a e
L
n xa f x dx
L L
4. Solution:
3. Determine modes:2
( ) sin , ( ) , 1, 2,3,...cn
tL
n
nx x v t e n
L
Solving Linear PDEs by Fourier Series
2. Separate variables: xx tk v ckv
6 0, ( ,0) is
reflectionlesst x xxxu uu u u x 1. KdV equation:
2
1
( , ) 4 ( , ),N
n n n nn
u x t k x t k
4. Solution by inverse scattering:
3. Determine spectrum: { , }n n
Solving Nonlinear PDEs by Inverse Scattering
2. Linearize KdV: ( , ) 0xx u x t
(discrete)
2
2
KdV: 6 0
Miura transformation:
mKdV: 6 0 (Burger type)
Cole-Hopf transformation:
Schroedinger's equation: ( , ) 0 (linear
t x xxx
x
t x xxx
x
xx
u uu u
u v v
v v v v
v
u x t
)
2. Linearize KdV
[ ( ,0) ] 0xx u x
Potential(t=0)
Eigenvalue(mode)
Eigenfunction
Schroedinger’s Equation(time-independent)
- Given a potential , determine the spectrum { , }.u
Scattering Problem:
- Given a spectrum { , }, determine the potential .u
Inverse Scattering Problem:
3. Determine Spectrum
1 2{0 ... }N (eigenvalues)
(eigenfunctions)1 2{ , ,..., }N
(a) Solve the scattering problem at t = 0 to obtainreflection-less spectrum:
(b) Use the fact that the KdV equation is isospectral to obtain spectrum for all t
1 2{ , ,..., }Nc c c (normalizing constants)
- Lax pair {L, A}: [ , ]
0
LL A
tt
At
(b) N-Solitons ([GGKM], [WT], 1970):
2
2( , ) 2 log det( )u x t I A
x
(a) Solve GLM integral equation (1955):
( ) 0 ( , ) 2 ( , , )xx u u x t K x x tx
4. Solution by Inverse Scattering
382( , ) n nk t k xnB x t c e
( , , ) ( , ) ( , ) ( , , ) 0x
K x y t B x y t B x z t K z y t dz
One-soliton (N=1):
1 1
2221
21
2 21 1 1
( , ) 2 log 12
2 sech
kcu x t e
x k
k k
Two-solitons (N=2):
1 1 2 2
1 1 2 2
2 222 21 2
21 2
2 2 22 21 2 1 2
1 2 1 2
( , ) 2 log 12 2
4
k k
k k
c cu x t e e
x k k
k k c ce
k k k k
Soliton matrix:
2, 4 (moving frame)m m n nk km nn n
m n
c cA e x k t
k k
Unique Properties of Solitons
Infinitely many conservation laws
Signature phase-shift due to collision
1
( , ) 4 nn
u x t dx k
(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:
Lasers:
- Temporal solitons (optical pulses)
- Spatial solitons (coherent beams of light)- BEC solitons (coherent beams of atoms)
Optical Phenomena
Hieu Nguyen:
Temporal solitons involve weak nonlinearity whereas spatial solitons involve strong nonlinearity
Hieu Nguyen:
Temporal solitons involve weak nonlinearity whereas spatial solitons involve strong nonlinearity Refraction Diffraction
Coherent Light
20t xxi
NLS Equation
2 2[ ( ) / 2 ( / 4) ]( , ) 2 sech[ ( )] i x t tx t x t e
Envelope
Oscillation
One-solitons:
Nonlinear termDispersion/diffraction term
Temporal Solitons (1980)
Chromatic dispersion:
Before After
- Pulse broadening effect
Self-phase modulation
- Pulse narrowing effect
Before After
Spatial Solitons
Diffraction
- Beam broadening effect:
Self-focusing intensive refraction (Kerr effect)
- Beam narrowing effect
BEC (1995)Cold atoms
http://cua.mit.edu/ketterle_group/
- Coherent matter waves- Dilute alkali gases
Atom Lasers
21
2t xxi
Gross-Pitaevskii equation:
Atom-atom interaction External potential
- Quantum field theory
Atom beam:
Molecular Lasers
2 2 *
2 2 2
1
21
( )4 2
t xx a am
t xx m am
i
i
(atoms)
(molecules)
Cold molecules
- Bound states between two atoms (Feshbach resonance)
Molecular laser equations:
Joint work with Hong Y. Ling (Rowan University)
Many Faces of Solitons
Quantum Field Theory
General Relativity
- Quantum solitons- Monopoles- Instantons
- 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
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 Vriesequation, 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/
References
www.rowan.edu/math/nguyen/soliton/