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/