K. MurawskiUMCS Lublin

Outline

• historical remarks - first observation of a soliton• definition of a soliton• classical evolutionary equations• IDs of solitons• solitons in solar coronal loops

Ubiquity of waves

John Scott Russell (1808-1882)

Union Canal at Hermiston, Scotland

- Scottish engineer at Edinburgh

First observation of Solitary Waves

“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.

Recreation of the Wave of Translation (1995)

Scott Russell Aqueduct on the Union Canal near Heriot-Watt University, 12 July 1995

Vph2 = g(h+h’)

J. Scott Russell experimented in the 30-foot tank which he built in his back garden in 1834:

???Oh

no!!!

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

Stationary Solutions

Steepen + Flatten = Stationary

- 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 (1-soliton):

- Assume wavelength approaches infinity

x

- u

x

- u

x

- u

x

- u

x

- u

Los Alamos, Summers 1953-4 Enrico Fermi, John Pasta, and Stan Ulamdecided to use the world’s then most powerful computer, the

MANIAC-1(Mathematical Analyzer Numerical Integrator And Computer)

to study the equipartition of energy expected from statistical mechanics in simplest classical model of a solid: a 1D chain of equal mass particles coupled by nonlinear* springs:

*They knew linear springs could not produce equipartition

Fixed = Nonlinear Spring fixed

NnNnnnn

1210

Fermi-Pasta-Ulam problem

M V(x)

V(x) = ½ kx2 + /3 x3 + /4 x4

1. Only lowest few modes (from N=64) excited.

What did FPU discover?

2. Recurrences

Note only modes 1-5

N-solitons

- Derived KdV eq. for the FPU system- Solved numerically KdV eq.- Solitary waves pass through each other- Coined the term ‘soliton’ (particle-like behavior)

Perring and Skyrme (1963)

Zabusky and Kruskal (1965):

Solitons and solitary waves -

definitions

A solitary wave is a wave that retains its shape, despite dispersion and nonlinearities.

A soliton is a pulse that can collide with another similar pulse and still retain its shape after the collision, again in the presence of both dispersion and nonlinearities.

Soliton collision: Vl = 3, Vs=1.5

Unique Properties of Solitons

Infinitely many conservation laws, e.g.

Signature phase-shift due to collision

1

( , ) 4 nn

u x t dx k

(conservation of mass)

vt + vxxx + 6v2vx= 0

mKdV solitons

modified Korteweg-de Vries equation

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

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 (1970):

2

2( , ) 2 log det( )u x t I A

x

(a) Solve Gelfand-Levitan-Marchenko 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

Other Analytical Methods of Solution

Hirota bilinear method

Backlund transformations

Wronskian technique

Zakharov-Shabat dressing method

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

- optical fibers

Breather soliton

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

Magnetic loops in solar corona (TRACE)

Strong B dominates plasma

       

TThin flux tubehin flux tube approximation approximation

• The dynamics of long wavelength (λ»a) waves may be described by the thin flux tube equations (Roberts & Webb, 1979; Spruit & Roberts, 1983 ).

V(z,t): longitudinal comp. of velocity

       

Model equationsModel equations

•Weakly nonlinear evolution of the waves is governed, in the cylindrical case, by the Leibovich-Roberts (LR) equation, viz.  

• and, in the case of the slab geometry, by the Benjamin - Ono (BO) equation, viz

•Roberts & Mangeney, 1982; Roberts, 1985

       

Algebraic solitonAlgebraic soliton

• The famous exact solution of the BO equation is the algebraic soliton,

• Exact analytical solutions of the LR equation have not been found yet!!!

       

MHD (auto)solitons in magnetic structuresMHD (auto)solitons in magnetic structures

• In presence of weak dissipation and active non-adiabaticity (e.g. when the plasma is weakly thermally unstable) equations LR and BO are modified to the extended LR or BO equations of the form  

• B: nonlinear, A:non-adiabatic, δ:dissipative and D:dispersive coefficients. It has been shown that when all these mechanisms for the wave evolution balance each other, equation eLR has autowave and autosoliton solutions.

• By definition, an autowave is a wave with the parameters (amplitude, wavelength and speed) independent of the initial excitation and prescribed by parameters of the medium only.

       

MHD (auto)solitons in magnetic structuresMHD (auto)solitons in magnetic structures

• For example, BO solitons with different initial amplitudes evolve to an autosoliton. If the soliton amplitude is less than the autosoliton amplitude, it is amplified, if greater it decays:

• The phenomenon of the autosoliton (and, in a more general case, autowaves) is an example of self-organization of MHD systems.

Ampflication dominates for larger and dissipation for shorter . Solitons with a small amplitude have larger length and are smoother than high amplitude solitons, which are shorter and steeper. Therefore, small amplitude solitons are subject to amplification rather than dissipation, while high amplitude solitons are subject to dissipation.

Solitons, Strait of Gibraltar

These subsurface internal waves occur at depths of about 100 m. A top layer of warm, relatively fresh water from the Atlantic Ocean flows eastward into the Mediterranean Sea. In return, a lower, colder, saltier layer of water flows westward into the North Atlantic ocean. A density boundary separates the layers at about 100 m depth.

Andaman Sea SolitonsAndaman Sea Solitons

Oceanic Solitons (Vance Brand Waves) are nonlinear, localized waves, that move in groups of six. They manifest as large internal waves, and move at a speed of 8 KPH. They were first recorded at depths of 120m by sensors on Oil Rigs in the Andaman Sea. Until that time Scientists denied their very existence…based on the fact that “There was no record of any such phenomenon.”

Oceanic Solitons (Vance Brand Waves) are nonlinear, localized waves, that move in groups of six. They manifest as large internal waves, and move at a speed of 8 KPH. They were first recorded at depths of 120m by sensors on Oil Rigs in the Andaman Sea. Until that time Scientists denied their very existence…based on the fact that “There was no record of any such phenomenon.”

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

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.

H. D. Nguyen, Decay of KdV Solitons, SIAM J. Applied Math. 63 (2003), No. 3, 874-888.

A. Snyder and F.Ladouceur, Light Guiding Light, Optics and Photonics News, February, 1999, p. 35

B. Seaman and H. Y. Ling, Feshbach Resonance and Coherent Molecular Beam Generation in a Matter Waveguide, preprint (2003).

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

References