the sine-gordon-equation - itp · 2017. 3. 6. · nonlinear optics k 1. seite 7 ... coupling of...

Institut für Theoretische Physik

The Sine-Gordon-Equation

Malte Misfeldt, Morten Pfeiffer 13.05.2015

Nonlinear ODE

Origin of the SGE

Transformation

Scope of applications

Chain of pendulums

Gaussian curvature

Solving the SGE

Case Analysis of the solutions

Solitons

Soliton Collision

Modern Science

Seite 2


Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik 13.05.2015

Nonlinear PDE

KdV (simplified):

Solving methods:

Linearization (e.g. small-angle-approximation)

Inverse scattering method

Numerical solutions (e.g. Newton‘s method)

)()()( vFuFvuF

vvufuvufvuuufu ttt )()(0)(

Seite 3



Origin of the SGE

Energy-momentum-relation:

Schrödinger-Equation with

Klein-Gordon-Equation (spinless particles!)

Klein-Gordon-Equation with any potential V

1 with222 cmpE

withˆt

iEE

H

i

m

tp with0

2

22

2

2

0V

0'22

2

V

t

Seite 4



Origin of the SGE

Sine-Gordon-Equation:

0 )sin( xxtt)()(V

sin' and with 2

2

Seite 5



Transformation

Transformation from

SGE reduces to

vutx , to ,

txv

txu

21

21

sinuv

vvuvvuuutt

vvuvvuuuxx

xv

vf

xu

uf

xf

41

41

0 )sin( xxtt

Seite 6



Scope of applications

Oscillation of a chain of pendulums

Differential geometry: surface with Gaussian curvature

Solid-state physics (crystals)

Tentative model of an elementary particle

Equivalent form of the Thirring model

Nonlinear optics

1K

Seite 7



Chain of pendulums

Seite 8



Chain of pendulums

GF MMI

SGE 0sin

, xX

:nKoordinate geeignete

I

cc GG

XXTT

tT

MdgcctI

MdgtI

GGxt

iiiiiit

| sin

)(sin

2

2

2

2

2

2

0

11

Seite 9



Gaussian curvature

surfaces with constant negative curvature

Coordinate transformation:

SGE! sin

1

1

cos

1

2

sin

2

2

222

222

uv

vuFEG

F

FEG

F

v

vu

u

uvuv FFK

xG

xxF

xE

dvGdudvFduEds

1K

Seite 10



Solving the SGE…

gkeitGeschwindi :v

enskonstantIntegratio :A

1sin22

sin1

,

2

212

2

vA

dz

v

vtxztx

zz

SGE

Seite 11



Case Analysis of the solutions

1. Periodical Solution, unstable

2. Periodical Solution, unstable

3. Screw-shaped Solution, stable

4. Screw-shaped Solution, stable

5. Soliton-Solution, unstable

6. Soliton-Solution, stable

1 ,20

1 ,20

2

2

vA

vA

1 ,2

1 ,0

2

2

vA

vA

1 ,0

1 ,2

2

2

vA

vA

(4 Subcases)

2

212 1sin22 vA

dz

Seite 12



Solitons

Stable, localized solitary wave (wave packet / pulse)

Definition is difficult to find but:

Solitons are of a permanent form (hump-shaped)

Solitons are localized within a region

Solitons emerge from the collision unchanged (except for a

phase shift)

E.g. water waves in a canal

or smoke rings

Seite 13



Soliton collision

Phase-shift (like KdV)

2

22

1cosh

1sinharctan4

vvt

vxvsol

Seite 14



Soliton collision

Bäcklund transformation (for SGE)

atug vf

d

du

d

v

u

axvu

vfaua

a

a

exparctan4,

tanlog22sin2

sin

sin

41

sin2

221

221

2

Seite 15



Soliton collision

2 solitons collide with 2 antisolitons

Seite 16



Breather

Coupling of kink and antikink

Special form of solitons

Exact solution via inverse scattering

Seite 17



Modern Science

Thirring Model

Equivalent to the quantum-SGE

Exactly solvable field equation

Josephson junction

Extremely rapid Switching element

Coupling:

Current by tunneling:

barrier , tx sincII

Seite 18



Modern Science

Einstein-Bose-condensate

Particle wave

Solitons in Rubidium-condensate (350 Atoms thickness)

Dispersion: non-linear Schrödinger-Equation

Dislocation in crystals

Crystals that oszillate due to heat and pressure

Mass and energy wave as kinks

Seite 19



Modern Science

Biophysics

DNA-proteins

Seperation of DNA

Double SGE:

Epilepsy

nerv pulse ↔ soliton

Medical: prognosis

22

22

cossinsin

cossinsin

xxtt

xxtt

Seite 20



Modern Science

Data transfer in glass fibers

Small amplitude of light Dispersion

: solitary wave

pulse duration: PicoSec TeraBits per Seconds

Labor: 180M km of stable solitons

m 3,1

Seite 21



Conclusion

Seite 22



Sources

http://lie.math.brocku.ca/~sanco/solitons/index.php

http://www.mleistner.de/images/wellen1.JPG

http://en.wikipedia.org/wiki/Breather, ~/Soliton, ~/Sine-Gordon_equation

P.G. Drazin: „Solitons“ (Cambridge University Press, 1983)

Attilio Maccari: „Nonlinear Field Equations and Solitons as Particles“ (Electronic journal of theoretical

physics, 2006)

Rainer Grauer, „Die Mechanik der Sine-Gordon-Solitonen“ (Heinrich-Heine-Universität Düsseldorf)

S.Aubry & P.Y.LeDaeron, „The Discrete Frenkel-Konkorova Model and its Extensions“, 1982

Isidoro Kimel, „On the Sine-Gordon-Thirring Model Equivalence at the classical level“, Sao Paulo,

1975

https://www.youtube.com/channel/UCkvAtOZ85ReX7H-Dc5WmxTA

Sidney Coleman, „Quantum-Sine-Gordon equation as the massive Thirring model“, Cambridge,

1975

David Gablinger, „Notes on the Sine-Gordon equation“, Hannover, 2007

http://www.nature.com/nature/journal/v474/n7353/images/nature10122-i1.0.jpg

the sine-gordon-equation - itp · 2017. 3. 6. · nonlinear optics k 1. seite 7 ... coupling of...

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