The Sine-Gordon-Equation

Transkript

1 The Sine-Gordon-Eqation Nonlinear ODE Origin of the SGE Transformation Scope of applications Chain of pendlms Gassian crvatre Solving the SGE Case Analysis of the soltions Solitons Soliton Collision Modern Science Malte Misfeldt, Morten Pfeiffer

2 Seite Institt für Theoretische Physik Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Nonlinear PDE KdV (simplified): Solving methods: Linearization (e.g. small-angle-approximation) Inverse scattering method Nmerical soltions (e.g. Newton s method) ) ( ) ( ) ( v F F v F v v f v f v f t t t ) ( ) ( 0 ) (

3 Origin of the SGE Energy-momentm-relation: E p m with c 1 Schrödinger-Eqation with ˆ H E with E i t V 0 Klein-Gordon-Eqation (spinless particles!) m t 0 with p i Klein-Gordon-Eqation with any potential V V' 0 t Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 3

4 Origin of the SGE Sine-Gordon-Eqation: tt sin( ) xx 0 with and V' ( ) sin( ) Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 4

5 Transformation tt sin( ) xx 0 Transformation from v 1 1 SGE redces to x t x t sin v x, t to, v f x xx tt f x f v v x v v v v vv vv Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 5

6 Scope of applications Oscillation of a chain of pendlms Differential geometry: srface with Gassian crvatre Solid-state physics (crystals) Tentative model of an elementary particle Eqivalent form of the Thirring model Nonlinear optics K 1 Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 6

7 Chain of pendlms Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 7

8 Seite 8 Institt für Theoretische Physik Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Chain of pendlms M F M G I SGE 0 sin, x X geeignetekoordinaten : I c c G G XX TT t T Mdg c c t I Mdg t I G G x t i i i i i i t sin ) sin( 0 1 1

9 Gassian crvatre srfaces with constant negative crvatre Coordinate transformation: ds E F G x x cos 1 E v v x x d 1 v 1 F ddv G dv vf F v K F vf EG F sin EGF sin SGE! K 1 Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 9

10 Solving the SGE SGE z x, t z x vt 1 v sin d zz A :Integrationskonstant e v :Geschwindi gkeit sin 1 1 A v Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 10

11 Case Analysis of the soltions z d 1. Periodical Soltion, nstable. Periodical Soltion, nstable 3. Screw-shaped Soltion, stable 4. Screw-shaped Soltion, stable 1 A sin 1 v 0 A, v 1 0 A, v 1 A 0, v 1 A, v 1 5. Soliton-Soltion, nstable 6. Soliton-Soltion, stable (4 Sbcases) A A 0, v, v 1 1 Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 11

12 Solitons Stable, localized solitary wave (wave packet / plse) Definition is difficlt to find bt: Solitons are of a permanent form (hmp-shaped) Solitons are localized within a region Solitons emerge from the collision nchanged (except for a phase shift) E.g. water waves in a canal or smoke rings Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 1

13 Soliton collision Phase-shift (like KdV) sol 4arctan v sinhx coshvt 1 v 1 v Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 13

14 Soliton collision Bäcklnd transformation (for SGE) 1 1 asin a sin v d d a sin d a log tan 1 sin f v t, v 4arctan g exp ax a f v 4 Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 14

15 Soliton collision solitons collide with antisolitons Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 15

16 Breather Copling of kink and antikink Special form of solitons Exact soltion via inverse scattering Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 16

17 Modern Science Thirring Model Eqivalent to the qantm-sge Exactly solvable field eqation Josephson jnction Extremely rapid Switching element barrier Copling: x, t Crrent by tnneling: I sin I c Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 17

18 Modern Science Einstein-Bose-condensate Particle wave Solitons in Rbidim-condensate (350 Atoms thickness) Dispersion: non-linear Schrödinger-Eqation Dislocation in crystals Crystals that oszillate de to heat and pressre Mass and energy wave as kinks Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 18

19 Modern Science Biophysics DNA-proteins Seperation of DNA Doble SGE: tt tt Epilepsy xx xx sin sin nerv plse soliton Medical: prognosis sin cos sin cos Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 19

20 Modern Science Data transfer in glass fibers Small amplitde of light Dispersion 1, 3m : solitary wave plse dration: PicoSec TeraBits per Seconds Labor: 180M km of stable solitons Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 0

21 Conclsion Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite 1

22 Sorces ~/Soliton, ~/Sine-Gordon_eqation P.G. Drazin: Solitons (Cambridge University Press, 1983) Attilio Maccari: Nonlinear Field Eqations and Solitons as Particles (Electronic jornal of theoretical physics, 006) Rainer Graer, Die Mechanik der Sine-Gordon-Solitonen (Heinrich-Heine-Universität Düsseldorf) S.Abry & P.Y.LeDaeron, The Discrete Frenkel-Konkorova Model and its Extensions, 198 Isidoro Kimel, On the Sine-Gordon-Thirring Model Eqivalence at the classical level, Sao Palo, Sidney Coleman, Qantm-Sine-Gordon eqation as the massive Thirring model, Cambridge, 1975 David Gablinger, Notes on the Sine-Gordon eqation, Hannover, Malte Misfeldt, Morten Pfeiffer Proseminar Theoretische Physik Seite