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Solitons around us. 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.

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Solitons around us

K. Murawski

UMCS Lublin


Outline
Outline

  • historical remarks - first observation of a soliton

  • definition of a soliton

  • classical evolutionary equations

  • IDs of solitons

  • solitons in solar coronal loops



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) rolling on at a rate of some

Scott Russell Aqueduct on the Union Canal

near Heriot-Watt University, 12 July 1995


J. Scott Russell rolling on at a rate of some experimented in the 30-foot tank

which he built in his back garden in 1834:

Vph2 = g(h+h’)


??? rolling on at a rate of some

Oh no!!!


Controversy Over Russell’s Work rolling on at a rate of some 1

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 rolling on at a rate of some

D.J. Korteweg and G. de Vries (1895)

  • surface elevation above equilibrium

  • depth of water

  • surface tension

  • density of water

  • force due to gravity

  • small arbitrary constant


Korteweg-de Vries (KdV) Equation rolling on at a rate of some

Rescaling:

KdV Equation:

Nonlinear Term

Dispersion Term

(Steepen)

(Flatten)


Sta rolling on at a rate of some tionary Solutions

Profile of solution curve:

  • Unchanging in shape

  • Bounded

  • Localized

Do such solutions exist?

Steepen + Flatten = Stationary


Solitary Wave Solutions rolling on at a rate of some

1. Assume traveling wave of the form:

2. KdV reduces to an integrable equation:

3. Cnoidal waves (periodic):


4. Solitary waves ( rolling on at a rate of some 1-soliton):

- Assume wavelength approaches infinity


M rolling on at a rate of some

V(x)

Fixed = Nonlinear Spring fixed

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

V(x) = ½ kx2 + /3 x3 + /4 x4


What did FPU discover? rolling on at a rate of some

Note only modes 1-5

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

  • Recurrences


N rolling on at a rate of some -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
Solitons and solitary waves rolling on at a rate of some - 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: V rolling on at a rate of some l = 3, Vs=1.5


Unique Properties of Solitons rolling on at a rate of some

Signature phase-shift due to collision

Infinitely many conservation laws, e.g.

(conservation of mass)


mKdV rolling on at a rate of some solitons

modified Korteweg-de Vries equation

vt + vxxx + 6v2vx= 0


Inverse Scattering rolling on at a rate of some

1. KdV equation:

2. Linearize KdV:

3. Determine spectrum:

(discrete)

4. Solution by inverse scattering:


2. Linearize KdV rolling on at a rate of some


Schroedinger’s Equation rolling on at a rate of some

(time-independent)

Potential

(t=0)

Eigenvalue

(mode)

Eigenfunction

Scattering Problem:

Inverse Scattering Problem:


3. Determine Spectrum rolling on at a rate of some

(a) Solve the scattering problem at t = 0 to obtain

reflection-less spectrum:

(eigenvalues)

(eigenfunctions)

(normalizing constants)

(b) Use the fact that the KdV equation is isospectral

to obtain spectrum for all t

- Lax pair {L, A}:


4. Solution by Inverse Scattering rolling on at a rate of some

(a) Solve Gelfand-Levitan-Marchenko integral equation (1955):

(b) N-Solitons (1970):


Soliton matrix: rolling on at a rate of some

One-soliton (N=1):

Two-solitons (N=2):


Other rolling on at a rate of some Analytical Methods of Solution

Hirota bilinear method

Backlund transformations

Wronskian technique

Zakharov-Shabat dressing method


Other Soliton Equations rolling on at a rate of some

Sine-Gordon Equation:

  • Superconductors (Josephson tunneling effect)

  • Relativistic field theories

Breather soliton

Nonlinear Schroedinger (NLS) Equation:

  • optical fibers


NLS Equation rolling on at a rate of some

Dispersion/diffraction term

Nonlinear term

One-solitons:

Envelope

Oscillation


Magnetic loops in solar corona rolling on at a rate of some (TRACE)

Strong B dominates plasma


T rolling on at a rate of some hin 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 rolling on at a rate of some

  • 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 rolling on at a rate of some

  • 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 rolling on at a rate of some

  • 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) independentof the initial excitation and prescribed by parameters of the medium only.


MHD (auto)solitons in magnetic structures rolling on at a rate of some

  • 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 rolling on at a rate of some

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 soliton s
Andaman Sea rolling on at a rate of some 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 rolling on at a rate of some

"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 rolling on at a rate of some

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


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