Lecture 3: Short Summary

Transcription

1 Lecture 3: Short Summary u t t u x x + s in u = 0 sine-gordon equation: Kink solution: Á = 4 arctan exp (x x0 ) Q= Topological charge: Light-cone coordinates: ¼ R x = Á = sin Á Bäcklund à Á sin Á+à à Á + sin Á Ã

2 sine-gordon model: -soliton interactions KK-collision Á = 4 arctan v=0.8 KK-collision v=0.8 = Á = 4 a rc t a n Breather: i! v=p! " " x v sinh p v vt cosh p v ; v= + sinh pv t v v cosh px v Á = 4 arctan "p ; # Á x t > 0 #! sin!t p! cosh! x Á # t x x t

3 sine-gordon model: -soliton interactions KK-collision sinh v t Á = 4 ar c t an ; = p v c os h x v vt ln v v t ln v e e = 4 tan e x + e x Á x t Asymptotic: t! ±t ±t Á ¼ ÁK (x + v t + + ÁK¹ (x v t Asymptotic: t! + ±t ±t Á ¼ ÁK (x + v t + ÁK¹ (x v t + The phase shift: p ln v ±t = = v ln v v

4 sine-gordon model: Lax pair formulation Recall: Lax pair is given by two linear equations ½ Ãxt = Lt à + LÃt ; Ãtx = Ax à + AÃx : sine-gordon: iutx Lt = ¾ ; Ãx = LÃ; Ãt = Aà Ã= Lt à + LAà = Ax à + ALÃ; à à à à Lt Ax = [A; L] Zero curvature condition i 0 ux i 0 L = i + = i ¾3 + ux ¾ ; C 0 ux 0 cos u 0 0 i cos u A= + = ¾3 + ¾ 4i 0 4i i 0 4i 4i Ax = ux sin u ¾3 + ux ¾ 4i 4i i i i [A; L] = ¾ ¾3 + sin u ¾ 4 4 in 0th order in λ sine-gordon equation is recovered! iutx i ¾ = sin u ¾

5 Interaction bertween the solitons -solitons solutions Á = ÁK (x d) + Ák (x + d) ¼ Linear perturbations of the kink: Á = ÁK (x) + (x; t) ÁK = 4 arctan ex (x; t) = <» (x)ei!t Eint (d) = EK K (d) M ¼ 3e d Homework: Prove (x; (x; t) + (x; t) cos ÁK (x) = 0 d +» (x) =!» (x) dx cosh x

6 Linearized perturbations of the sg kink d +» ( x ) =!» (x) dx cosh x d d y a ^a ^»(x) =!»(x); a ^= + tanh x; a ^ = + tanh x dx dx d ^»0 + tanh x» (x) = 0 Vacuum state: a» ( x ) = 0 dx cosh x y Zero mode C C d Ák C Á = Á K ( x ) + C» 0 ( x ) = 4 a rc t a n e x + = ÁK + ¼ ÁK x + c o sh x dx»k (x) = (tanh x + ik)e ; d + tanh x eikx =»k (x) dx Continuum modes: Note: a ^ eikx ikx» ( ) = ( + ik )eikx ; p!k = + k Reflectionless Reflectionlesspotential! potential!» () = ( + ik )ei(kx+±) ; ei± = ik + ik

7 sine-gordon massive Thirring model S= Z d Á@ ¹ Á ( cos Á) Thirring model Invariancies: Bosonization: S= Á! Á0 = Á + Z sine-gordon model h i g ¹ ¹ d x iã¹ Ã + mã¹ã (ù ¹ à )(ù à ) ¼ n ; 0 = ¾ ; = i¾ ; 5 = 0 = ¾3 Ã! Ã0 = ei V Ã; Ã! Ã0 = ei 5 A à 5 iá ¹ mã à = e Meson states fermion-anti fermion bound states 4¼ = +g =¼ (S.Coleman, 975) Soliton fundamental fermion º The topological current of the sine-gordon model J¹ = ¼ Á ¹ coincides with the Noether current of the massive Thirring model j¹ = iã¹ Ã

8 Solitons vs. Solitary Waves Equation S-G: ÁÄ Á00 + sin Á = 0 λ 4 : ÁÄ Á00 Á + Á3 = 0 Solution YES NO! ÁK K¹ = 4 arctan (e x+x0 ) ÁK K¹ = a tanh ³ m(x x0 ) p How do we know if it is integrable or it is a non-integrable? Historically, combination of experimental mathematics ( 4) and known analytic solutions (S-G), then inverse scattering transform, group theoretic structure (Kac-Moody Algebras), Painlevé test. Does any part of hierarchy of solitons in integrable theories (S-G breather) exist in non-intergrable theories?

9 Topology primer: maps and windings Kinks in d: + Space: - Vacuum: + Maps: - Topological charge: Q = R = Á() Á( ) Circles: S S Space: Vacuum: Á = (sin '; cos ') Maps:

10 Circles: S S Topological charge: Z¼ d Á Q = ¼ d' " Á d' 0 Vacuum: Q=0: Á = (0; ) Q=: Á = (sin '; cos ') Q=: Á = (sin '; cos ')

11 Scaling agruments: Derrick s theorem Consider a model with scalar field in d-dim R d E [Á] = d x [@¹ Á@ ¹ Á + U (Á)] = E + E0 Scale transformation: ~x! ~y = Á(~x) = dd x! dd ( x) d = d dd Á(~ Á( ( x¹~x)) E [Á]! d E + d E0 Each term is positive. If there is a stationary point of E(λ)? de [ Á] d = ( d) d E d d E0 d= d= d=3

12 For a simple model E [Á] = R dd x [@¹ Á@ ¹ Á + U (Á)] = E + E0 nontrivial solutions (E 0, E0 0 ) are possible only in d= There are 3 possibilities to evade Derrick s theorem: d=: take E0 = 0, then the model is scale-invariant Extend the model including higher derivatives in ϕ (Skyrme model in d=3, baby Skyrme model in d=, Faddeev-Skyrme model in d=3) Extend the model including gauge fields (monopoles in d=3, instantons in Euclidean space d=4) ~x! ~x = ~y ; A¹ (~x)! A¹ (~y ); E [Á] = R D¹ Á(~x)! D¹ Á(~y); F¹º (~x)! F¹º (~y ) d x jf¹º j + jd¹ Áj + U (Á) = E4 + E + E0 dd x E [Á]! 4 d E4 + d E + d E0

13 If we restrict ourselves to the models with quadratic terms in derivatives, there are possibilities: d=: there are soliton solutions in the models with gauge and scalar fields or in pure scalar models with a potential U(ϕ) (Kinks). d=: there are soliton solutions in the models with gauge and scalar fields (vortices) or in pure scalar models without potential U(ϕ) (Lumps). d=3: there are soliton solutions in the models with gauge and scalar fields (monopoles) d=4: there are soliton solutions in the models with gauge field only (instantons) d>4: there are no soliton solutions, higher derivatives are necessary. Alternative: one can consider time-dependent stationary configurations!

14 4 model L= Á U (Á); ¹ Field equation: U (Á) = ¹ Á + R Potential energy: V = dx Kinetic energy: T = Vacuum: Á0 = a 4 Á a R t +U (Á) Static configuration: T=0 Energy bound: E = V = R i p R p 0 dx p Á U (Á) dx U (Á) Á0 0 h

15 4 model: Applications Phenomenological theory of second order phase transitions A model of the displacive phase transitions A model of uniaxial ferroelectrics A phenomenological theory of the non-perturbative transition in polyacetylene chain Condensed matter physics: solitary waves in shape -memory alloys Cosmology: model dynamics of the domain walls. Biophysics: soliton excitations in DNA double helices. Quantum field theory: a model example to investigate transition between perturbative and non-perturbative sectors of the theory. A model of quantum mechanical instanton transitions in doublewell potential

16 4 model: Kink solutions U (Á) = (Á ) ; # Z V = dx + (Á Minimum of the energy: = ( Á kink solution: ÁK = tanh(x x0 ); Energy density: E = x x0 = Z dá = t a n h Á Á ÁK¹ = tanh(x x0 ) cosh4 (x x0 ) Mass of the kink: M= Z E dx = Topological charge: Q= = Topological current: J¹ = [Á() Á( )] º Á; ¹ J¹ 0 4 3

17 Interaction between the kinks Kink-antikink pair (a=, m = ): Á(x) = + tanh(x R) tanh(x + R) Far away from the pair (somewhere at x 0) tanh(x R) ¼ + e(x R) ; tanh(x + R) ¼ e (x+r) Interaction energy: Eint ¼ 6e L ; L = R Kinks attracts each other with the force F = deint dl Linear oscillations on the static kink background: ÁÄ Á00 (a Á )Á = 0 ¼ 3e L Á = ÁK + ± Á h i ± ÁÄ ± Á cosh6 x) ± Á = 0

18 Linearized perturbations of the 4 kink d 6 +4» =!» dx cosh x Reflectionless Reflectionlesspotential, potential,again! again! Homework: Prove it! n= a ^ya ^»(x) =!»(x); a ^= d d + n tanh x; a ^y = + n tanh x dx dx [a ^y ; a ^] = Vacuum state: a ^»0 n cosh x d (n) + n tanh x»0 (x) = 0 dx Internal mode: Continuum: sinh x a ^»0 =» = cosh x y ikx»k = e (n)»0(n)(x) = coshn x p! = 3 3 tanh x 3ik tanh x k Zero mode

19 Linearized perturbations of the 4 kink d 6 +4» =!» dx cosh x (n)»0 (x) = ; coshn x!0 = 0 Internal mode: sinh x a ^»0 =» = ; cosh x y p! = 3 Zero mode:

20 4 model: continuum modes:!k = 4+k ; Coupling R ikx»k (x) = < e 3 tanh x 3ik tanh x k dx k 0 (negative radiation pressure) The 4 kink accelerates towards the source of the radiation

21 Oscillon state: 4 model (I. L. Bogolubsky and V. G. Makhankov (JETP Lett. 4, (976)) In the 4 model there is a long lived nonradiative spatially localized solution (at least 0 million oscillations!!) oscillations Gaussian initial data: Á(x; 0) = 0:7e 0:05x Collective coordinate model: ³ xx Á(x; t) = A(t)e (A_ ) L=x0 = (A) 4 3A ³ ¼A 4 + ¼ A3 3x0 A A 0 Anharmonic oscillator with q frequency 0= 4 + 3x0

22 4 Kink-oscillon collisions vin = 0:

23 4 Kink-oscillon collisions vin = 0:

24 Sine-Gordon kink-breather collision vin = 0:5

25 4 KK collisions: fractal dynamics M. J. Ablowitz, M. D. Kruskal and J. F. Ladik (SIAM J.Appl. Math. 36 (979) 4) D. Campbell, J. Schonfeld and C Wingate (Physica 9D (983) ) P. Anninos, S. Oliveira and R. A. Matzner (Phys. Rev. D 44 (99) 47) etc Annihilation: ¹! oscillon KK vin = 0:7

26 4 KK collisions: fractal dynamics Bounce: ¹! KK ¹ KK vin = 0:7

27 4 KK collisions: fractal dynamics Three bounce resonance: ¹! KK ¹ KK vin = 0:4385

28 The resonance mechnism Kink-antikink collisions on x [ ; ]: Resonant energy exchange between the translational and the internal modes: the first collision excites the internal mode which takes the kinetic energy of the kinks, the second collision unbinds the pair taking the stored energy back: T = + n (D. Campbell, J. Schonfeld and C Wingate Physica 9D (983) ) The energy can be stored not only in the internal mode of the kink, but also in the collective modes

29 Boundary 4 model L= Á@ Á Á Neumann boundary Á(0; t) = H Boundary magnetic field - x 0 Kink solution on x [ ; ] : ÁK K¹ = tanh (x x0 ) Boundary energy: H Á(0; t) M= 4 3

30 Boundary 4 model: Energy functional E [Á] = Z0 E[Á ] = 0 3 dx[áx (Á )] [ Á Á] H Á(0; t) 3 ( H )3= ; 3 3 Kink-boundary forces: E [Á ] = + ( H )3= ; 3 3 F = 3 E[Á3 ] = H + ex0 ex0 4 repulsion repulsionfar farfrom fromthe theboundary boundary and andattraction attractionnear nearitit ( + H )3= 3 3 x0 < 0

31 H=-0.5 elastic recoil

32 Boundary 4 model: Static solutions H=0 bounces + excitations

33 Boundary 4 model: phase diagram vcr = q 4[( + H )3= + ( H )3= ]