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					               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:       3 g           x             2
              t      t, x     ,   2 u  
                 2 l                       3
                                                          u
KdV Equation:                                        ut 
                     ut  6uu x  u xxx  0               t
                                                          u
                                                     ux 
         Nonlinear Term           Dispersion Term         x
          ut  6uux  0              ut  uxxx  0




             (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:
              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      
        U ( z )   sech 
                        2
                              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                            Frequency domain
Fourier Series:
                            
                                  n x          n x 
         f ( x)  a0    an cos       bn sin      
                       n 1       L             L 




                        4          1          1               
             f ( x)       sin  x  sin 3 x  sin 5 x  ... 
                        
                                   3          5               

     http://mathworld.wolfram.com/FourierSeriesSquareWave.html
       Solving Linear PDEs by Fourier Series
                                           u (0, t )  u ( L, t )  0
1. Heat equation:       ut  c u xx ,
                               2

                                              u ( x, 0)  f ( x)

2. Separate variables:  xx  k                    vt  ckv

3. Determine modes:
                                                     cn 
                                                           2

                             n                           t
               n ( x)  sin    x,      v(t )  e    L 
                                                                ,   n  1, 2,3,...
                              L
                                                     cn 
                                                           2
                               
                                          n x            t
4. Solution:        u ( x, t )   an sin      e     L 

                                  n 1      L
                             2 L           n x
                    an   f ( x) sin           dx
                             L  0           L
Solving Nonlinear PDEs by Inverse Scattering

1. KdV equation:        ut  6uux  uxxx  0, u( x,0) is
                                                  reflectionless
2. Linearize KdV:       xx  u( x, t )  0

3. Determine spectrum:       {n , n }    (discrete)

4. Solution by inverse scattering:
                               N
               u ( x, t )  4 knn2 ( x, t ),   kn  n
                              n 1
                   2. Linearize KdV
KdV:                       ut  6uu x  u xxx  0
                                 
Miura transformation:           u  v 2  vx
                                 
mKdV:                    vt  6v 2 vx  vxxx  0    (Burger type)
                                 
                                     x
Cole-Hopf transformation:         v
                                     
                               
Schroedinger's equation:  xx  u ( x, t )  0     (linear)
           Schroedinger’s Equation
                   (time-independent)

                xx  [u( x,0)   ]  0

           Potential       Eigenvalue       Eigenfunction
            (t=0)            (mode)

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
                               L
                                   [ L, A]
                               t             
     - Lax pair {L, A}:                         0
                                            t
                                   A
                               t
         4. Solution by Inverse Scattering
(a) Solve GLM integral equation (1955):

   B( x, t )   c e2 8 kn t  k n x
                         3

                    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           ,    n  x  4kn2t (moving frame)
         km  kn             
One-soliton (N=1):
                                2         c12 2 k11 
                 u ( x, t )  2 2 log 1      e 
                                x      2k1           
                         2k12sech 2  k1 1   
Two-solitons (N=2):
                   2                      2
                               c12 2 k11 c2 2 k2 2
    u ( x, t )  2 2 log 1     e            e    
                   x      2k1            2k 2

                                k1  k2  c12c2 2 k11 2 k2 2 
                                            2
                                               2
                                              e               
                                k1  k2  4k1k2                 
                                                                 
               Unique Properties of Solitons

Signature phase-shift due to collision




Infinitely many conservation laws
                           

           u ( x, t )dx  4 kn   (conservation of mass)
     
                           n 1
      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                Optical Phenomena
whereas spatial
solitons involve
strong
nonlinearity        Refraction                    Diffraction




                                 Coherent Light
                   NLS Equation
                   it   xx     0
                                 2




Dispersion/diffraction term                    Nonlinear term


One-solitons:
                                           i [ ( x  t ) / 2 ( 2  2 / 4) t ]
     ( x, t )  2 sech[ ( x   t )]e

                                                           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
         it    xx      
                           2

                2


 Atom-atom interaction         External potential
                    Molecular Lasers

 Cold molecules

    - Bound states between two atoms (Feshbach resonance)
  Molecular laser equations:
        1
                            
it    xx  a   am       *
        2
                    2      2
                                                 (atoms)
        1
                                          
i t    xx  m   am    (   )   2 (molecules)
        4
                      2      2

                                            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/

				
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