SOLITONS
From Canal Water Waves to Molecular Lasers
Hieu D. Nguyen
Rowan University
IEEE Night
5-20-03
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…
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·ton
Pronunciation: 'sä-l&-"tän
Function: noun
Etymology: solitary + 2-on
Date: 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)
 3 g   1 2 2
1  2 

      2 
t 2 l x  2
3
3 x 
 - surface elevation above equilibrium
l - depth of water
T - surface tension
 - density of water
g - force due to gravity
 - small arbitrary constant
1 3 Tl
 l 
3
g
Korteweg-de Vries (KdV) Equation
Rescaling:
t
KdV Equation:
3 g
x
2
t, x  
,   2 u  
2 l
3

ut  6uu x  u xxx  0
Nonlinear Term
ut  6uux  0
Dispersion Term
ut  uxxx  0
(Steepen)
(Flatten)
u
t
u
ux 
x
ut 
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:
u ( x, t )  U ( z ),
z  x  ct
2. KdV reduces to an integrable equation:
dU
dU d 3U
c
 6U
 3 0
dz
dz
dz
3. Cnoidal waves (periodic):
U ( z )  a cn 2  bz   , k 
4. Solitary waves (one-solitons):
- Assume wavelength approaches infinity
 c

c
2
U ( z )   sech 
z  
2
 2

u ( x, t )  2k 2sech 2  k ( x  4k 2t )   )  , c  4k 2
-u
x
Other Soliton Equations
Sine-Gordon Equation:
uxx  utt  sin u
- Superconductors (Josephson tunneling effect)
- Relativistic field theories
Nonlinear Schroedinger (NLS) Equation:
iut  u u  u xx  0
2
- 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
Fourier Series:
Frequency domain
n x
n x 

f ( x)  a0    an cos
 bn sin

L
L

n 1 
f ( x) 

4
1
1

sin

x

sin
3

x

sin
5

x

...

 
3
5

http://mathworld.wolfram.com/FourierSeriesSquareWave.html
Solving Linear PDEs by Fourier Series
ut  c u xx ,
2
1. Heat equation:
u (0, t )  u ( L, t )  0
u ( x, 0)  f ( x)
2. Separate variables:  xx  k
vt  ckv
3. Determine modes:
n ( x)  sin
4. Solution:
n
x,
L
 cn 

 t
 L 
2
v(t )  e
 cn 

 t
 L 
n x
u ( x, t )   an sin
e
L
n 1
2 L
n x
an   f ( x)sin
dx
0
L
L

,
2
n  1, 2,3,...
Solving Nonlinear PDEs by Inverse Scattering
1. KdV equation:
2. Linearize KdV:
ut  6uux  uxxx  0, u ( x, 0) is
 xx  u ( x, t )  0
3. Determine spectrum:
{n , n }
reflectionless
(discrete)
4. Solution by inverse scattering:
N
u ( x, t )  4 knn2 ( x, t ),
n 1
kn  n
2. Linearize KdV
KdV:
ut  6uu x  u xxx  0

u  v 2  vx
Miura transformation:

mKdV:
vt  6v 2 vx  vxxx  0
(Burger type)

Cole-Hopf transformation:
x
v


Schroedinger's equation:  xx  u ( x, t )  0
(linear)
Schroedinger’s Equation
(time-independent)
 xx  [u ( x,0)   ]  0
Potential
(t=0)
Eigenvalue
(mode)
Eigenfunction
Scattering Problem:
- Given a potential u, determine the spectrum { , }.
Inverse Scattering Problem:
- Given a spectrum { , }, determine the potential u.
3. Determine Spectrum
(a) Solve the scattering problem at t = 0 to obtain
reflection-less spectrum:
{0  1  2  ...  N }
(eigenvalues)
{1 , 2 ,...,  N }
(eigenfunctions)
{c1 , c2 ,..., cN }
(normalizing constants)
(b) Use the fact that the KdV equation is isospectral
to obtain spectrum for all t
- Lax pair {L, A}:
L
 [ L, A]

t

0

t
 A
t
4. Solution by Inverse Scattering
(a) Solve GLM integral equation (1955):
B( x, t )   c e
2 8 kn3t  kn x
n

K ( x, y, t )  B( x  y, t )   B( x  z, t ) K ( z, y, t )dz  0
x
 xx  (u   )  0 

u ( x, t )  2 K ( x, x, t )
x
(b) N-Solitons ([GGKM], [WT], 1970):
2
u ( x, t )  2 2 log det( I  A)
x
Soliton matrix:
 cm cn km m  kn n 
A
e
,
 km  kn

 n  x  4kn2t (moving frame)
One-soliton (N=1):

c12 2 k11 
2
u ( x, t )  2 2 log 1 
e 
x
 2k1

 2k12sech 2  k1 1   
Two-solitons (N=2):

c12 2 k11 c22 2 k2 2
2
u ( x, t )  2 2 log 1 
e

e

x
2k 2
 2k1
 k1  k2  c12 c22 2 k11 2 k2 2 


e


 k1  k2  4k1k2

2
Unique Properties of Solitons
Signature phase-shift due to collision
Infinitely many conservation laws




u ( x, t )dx  4 kn
n 1
(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
NLS Equation
it   xx     0
2
Dispersion/diffraction term
Nonlinear term
One-solitons:
 ( x, t )  2 sech[( x   t )]e
i[ ( x  t ) / 2( 2  2 / 4) t ]
Envelope
Oscillation
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
BEC (1995)
Cold atoms
- Coherent matter waves
- Dilute alkali gases
http://cua.mit.edu/ketterle_group/
Atom Lasers
Atom beam:
Gross-Pitaevskii equation:
- Quantum field theory
1
2
it    xx      
2
Atom-atom interaction
External potential
Molecular Lasers
Cold molecules
- Bound states between two atoms (Feshbach resonance)
Molecular laser equations:


1
2
2
(atoms)
it    xx  a   am       *
2
1

2
2
i t    xx  m   am    (   )   2 (molecules)
4
2


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/