K. Murawski
UMCS 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
First observation of Solitary Waves
John Scott Russell (1808-1882)
- Scottish engineer at Edinburgh
Union Canal at Hermiston, Scotland
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.
Recreation of the Wave of Translation (1995)
Scott Russell Aqueduct on the Union Canal
near Heriot-Watt University, 12 July 1995
J. Scott Russell experimented in the 30-foot tank
which he built in his back garden in 1834:
Vph2 = g(h+h’)
???
Oh
no!!!
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 
Stationary Solutions
Profile of solution curve:
- Unchanging in shape
- Bounded
- Localized
Do such solutions exist?
Steepen + Flatten = Stationary
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 (1-soliton):
- Assume wavelength approaches infinity
u ( x, t )  2k 2sech 2  k ( x  4k 2t )   )  , c  4k 2
-u
x
Fermi-Pasta-Ulam problem
Los Alamos, Summers 1953-4 Enrico Fermi, John Pasta, and Stan Ulam
decided 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
M



n  0 n 1 n  2
Fixed



n  N 1 n  N
= Nonlinear Spring
V(x) = ½ kx2 + /3 x3 + /4 x4
fixed
V(x)
What did FPU discover?
1.
Only lowest few modes (from N=64) excited. Note only modes 1-5
2.
Recurrences
N-solitons
Perring and Skyrme (1963)
Zabusky and Kruskal (1965):
-
Derived KdV eq. for the FPU system
Solved numerically KdV eq.
Solitary waves pass through each other
Coined the term ‘soliton’ (particle-like behavior)
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
Signature phase-shift due to collision
Infinitely many conservation laws, e.g.




u ( x, t )dx  4 kn
n 1
(conservation of mass)
mKdV solitons
modified Korteweg-de Vries equation
vt + vxxx + 6v2vx= 0
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 Gelfand-Levitan-Marchenko 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 (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
Other Analytical Methods of Solution
Hirota bilinear method
Backlund transformations
Wronskian technique
Zakharov-Shabat dressing method
Other Soliton Equations
Sine-Gordon Equation:
uxx  utt  sin u
- Superconductors (Josephson tunneling effect)
- Relativistic field theories
Breather soliton
Nonlinear Schroedinger (NLS) Equation:
iut  u u  u xx  0
2
- optical fibers
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
Magnetic loops in solar corona (TRACE)
Strong B dominates
plasma
Thin flux tube 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 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 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 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 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:
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.
• The phenomenon of the autosoliton (and, in a more general case, autowaves)
is an example of self-organization of MHD systems.
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 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.”
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
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.
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 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