Dimer with gain and loss: Integrability and PT -symmetry restoration

Transcription

1 Dimer with gain and loss: Integrability and PT -symmetry restoration I Barashenkov, University of Cape Town Joint work with: Dima Pelinovsky (McMaster University, Ontario) Philippe Dubard (University of Cape Town, Western Cape) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

2 Dimer with gain and loss: Integrability and PT -symmetry restoration I Barashenkov, University of Cape Town Joint work with: Dima Pelinovsky (McMaster University, Ontario) Philippe Dubard (University of Cape Town, Western Cape) PT -symmetric gain-loss extensions of conservative dimers define completely integrable Hamiltonian systems I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

3 Dimer with gain and loss: Integrability and PT -symmetry restoration I Barashenkov, University of Cape Town Joint work with: Dima Pelinovsky (McMaster University, Ontario) Philippe Dubard (University of Cape Town, Western Cape) PT -symmetric gain-loss extensions of conservative dimers define completely integrable Hamiltonian systems Selected nonlinearities produce PT -symmetry restoration I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

4 The nonlinear Schrödinger dimer with F and G cubics. i u + v = F(u, u, v, v ) i v + u = G(u, u, v, v ), I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

5 The nonlinear Schrödinger dimer with F and G cubics. i u + v = F(u, u, v, v ) i v + u = G(u, u, v, v ), Invariance w.r.t. time translations of the underlying oscillatory system, x(τ) = u(t)e iωτ + c.c. +..., y(τ) = v(t)e iωτ + c.c. +..., dictates U(1) phase invariance: F(e iφ u, e iφ u, e iφ v, e iφ v ) =e iφ F(u, u, v, v ), G(e iφ u, e iφ u, e iφ v, e iφ v ) =e iφ G(u, u, v, v ) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

6 Left-right symmetry and Hamiltonian formulation i u + v = F(u, u, v, v ) i v + u = G(u, u, v, v ) Left-right parity: F(u, u, v, v ) = G(v, v, u, u ) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

7 Left-right symmetry and Hamiltonian formulation i u + v = F(u, u, v, v ) i v + u = G(u, u, v, v ) Left-right parity: F(u, u, v, v ) = G(v, v, u, u ) Energy conservation gives rise to the Hamiltonian structure: i du dt = H u, i dv dt = H v (straight gradient) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

8 Left-right symmetry and Hamiltonian formulation i u + v = F(u, u, v, v ) i v + u = G(u, u, v, v ) Left-right parity: F(u, u, v, v ) = G(v, v, u, u ) Energy conservation gives rise to the Hamiltonian structure: i du dt = H u, i dv dt = H v (straight gradient) or i du dt = H v, i dv dt = H u (cross gradient) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

9 Hamiltonian + U(1)-invariant + left-right symmetric General straight-grad dimer: where i du dt = H u, i dv dt = H v, H = (uv + u v) + W (u, v), W = β 1 ( u 2 + v 2 ) 2 +β 2 u 2 v 2 +β 3 (u v+uv )( u 2 + v 2 )+β 4 (u v+uv ) 2 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

10 Hamiltonian + U(1)-invariant + left-right symmetric General straight-grad dimer: where i du dt = H u, i dv dt = H v, H = (uv + u v) + W (u, v), W = β 1 ( u 2 + v 2 ) 2 +β 2 u 2 v 2 +β 3 (u v+uv )( u 2 + v 2 )+β 4 (u v+uv ) 2 General cross-grad dimer: where i du dt = H v, i dv dt = H u, H = ( u 2 + v 2 ) + W (u, v), W = α 1 ( u 2 + v 2 ) 2 +α 2 u 2 v 2 +α 3 (u v+uv )( u 2 + v 2 )+α 4 (u v+uv ) 2 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

11 From left-right invariance to PT -symmetry i u + v = F(u, u, v, v ), i v + u = G(u, u, v, v ). Assume the left-right symmetry lifted: F(u, u, v, v ) G(v, v, u, u ), I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

12 From left-right invariance to PT -symmetry i u + v = F(u, u, v, v ), i v + u = G(u, u, v, v ). Assume the left-right symmetry lifted: F(u, u, v, v ) G(v, v, u, u ), e.g. by gain and loss: I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

13 From left-right invariance to PT -symmetry i u + v = F(u, u, v, v ), i v + u = G(u, u, v, v ). Assume the left-right symmetry lifted: F(u, u, v, v ) G(v, v, u, u ), e.g. by gain and loss: The case of generalised parity invariance the PT symmetry: ( ) ( ) u v P =, T u(t) = u ( t), T v(t) = v ( t) v u I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

14 Cross-gradient PT -symmetric dimer i du dt = H v, i dv dt = H u, where H = ( u 2 + v 2 ) + W (u, v) + iγ(uv u v), W = α 1 ( u 2 + v 2 ) 2 +α 2 u 2 v 2 +α 3 (u v+uv )( u 2 + v 2 )+α 4 (u v+uv ) 2 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

15 Cross-gradient PT -symmetric dimer i du dt = H v, i dv dt = H u, where H = ( u 2 + v 2 ) + W (u, v) + iγ(uv u v), W = α 1 ( u 2 + v 2 ) 2 +α 2 u 2 v 2 +α 3 (u v+uv )( u 2 + v 2 )+α 4 (u v+uv ) 2 A 4-parameter family of cross-gradient PT -symmetric dimers: i u + v iγu = α 3 ( u v 2 )u + α 3 v 2 u [ +2α 4 u 2 v + (2α 1 + α 2 + 2α 4 ) u 2 + 2α 1 v 2] v i v + u + iγv = α 3 (2 u 2 + v 2 )v + α 3 u 2 v [ +2α 4 v 2 u + (2α 1 + α 2 + 2α 4 ) v 2 + 2α 1 u 2] u I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

16 Cross-gradient PT -symmetric dimer i du dt = H v, i dv dt = H u, where H = ( u 2 + v 2 ) + W (u, v) + iγ(uv u v), W = α 1 ( u 2 + v 2 ) 2 +α 2 u 2 v 2 +α 3 (u v+uv )( u 2 + v 2 )+α 4 (u v+uv ) 2 Example 1: α 3 = 1, α 1 = α 2 = α 4 = 0. BE condensate in double-well potential; u = symmetric mode, v =antisymmetric mode i u + v iγu = ( u v 2 )u + v 2 u i v + u + iγv = (2 u 2 + v 2 )v + u 2 v ( IB & M Gianfreda 2014 J Phys A: Math Theor 47, ) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

17 Cross-gradient PT -symmetric dimer i du dt = H v, i dv dt = H u, where H = ( u 2 + v 2 ) + W (u, v) + iγ(uv u v), W = α 1 ( u 2 + v 2 ) 2 +α 2 u 2 v 2 +α 3 (u v+uv )( u 2 + v 2 )+α 4 (u v+uv ) 2 Example 2: α 2 = α 3 = 0; α 1 = α 4 = 1/2 (two cubic oscillators with periodically modulated coupling) i u + v iγu = u 2 v + (2 u 2 + v 2 )v i v + u + iγv = v 2 u + (2 v 2 + u 2 )u I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

18 Cross-grad PT dimer: Solutions Stokes vector : Z = u v +v u, Y = i(u v v u), X = u 2 v 2 : I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

19 Cross-grad PT dimer: Solutions Stokes vector : Z = u v +v u, Y = i(u v v u), X = u 2 v 2 : Ż = 0 Ẏ = 2X + 2α 3 XZ + 4α 1 XR Ẋ = 2Y + 2γR 2α 3 YZ (α 2 + 4α 1 )Y R. I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

20 Cross-grad PT dimer: Solutions Stokes vector : Z = u v +v u, Y = i(u v v u), X = u 2 v 2 : Ż = 0 Ẏ = 2X + 2α 3 XZ + 4α 1 XR Ẋ = 2Y + 2γR 2α 3 YZ (α 2 + 4α 1 )Y R. I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

21 Cross-grad PT dimer: Solutions Stokes vector : Z = u v +v u, Y = i(u v v u), X = u 2 v 2 : Ż = 0 Ẏ = 2X + 2α 3 XZ + 4α 1 XR Ẋ = 2Y + 2γR 2α 3 YZ (α 2 + 4α 1 )Y R. pic from: IB & M Gianfreda J Phys A 47 (2014) α 2 Trajectories: Z = Z 0, 4 Y 2 + α 3 Z 0 R + α 1 R 2 (γy + R) = C, where R = X 2 + Y 2 + Z0 2 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

22 Straight-gradient PT -symmetric dimer General conservative straight-grad dimer (no gain-loss yet): where i du dt = H u, i dv dt = H v, H = (uv + u v) + W (u, v), W = β 1 ( u 2 + v 2 ) 2 +β 2 u 2 v 2 +β 3 (u v+uv )( u 2 + v 2 )+β 4 (u v+uv ) 2 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

23 Straight-gradient PT -symmetric dimer General conservative straight-grad dimer (no gain-loss yet): where i du dt = H u, i dv dt = H v, H = (uv + u v) + W (u, v), W = β 1 ( u 2 + v 2 ) 2 +β 2 u 2 v 2 +β 3 (u v+uv )( u 2 + v 2 )+β 4 (u v+uv ) 2 The PT -symmetric extension: i du dt iγu = H u, i dv dt + iγv = H v. I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

24 Straight-gradient PT -symmetric dimer General conservative straight-grad dimer (no gain-loss yet): where i du dt = H u, i dv dt = H v, H = (uv + u v) + W (u, v), W = β 1 ( u 2 + v 2 ) 2 +β 2 u 2 v 2 +β 3 (u v+uv )( u 2 + v 2 )+β 4 (u v+uv ) 2 The PT -symmetric extension: i du dt iγu = H u, i dv dt + iγv = H v. Example: the standard" dimer i u + v + u 2 u = iγu i v + u + v 2 v = iγv I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

25 Straight-grad PT dimer: Solution in the Stokes variables Ẋ = β 2 YZ Ẏ = 2Z + (β 2 + 4β 4 )XZ + 2β 3 Z R Ż = 2Y + 2γR 4β 4 XY 2β 3 RY I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

26 Straight-grad PT dimer: Solution in the Stokes variables Divide by Ṙ = 2γZ : Ẋ = β 2 YZ Ẏ = 2Z + (β 2 + 4β 4 )XZ + 2β 3 Z R Ż = 2Y + 2γR 4β 4 XY 2β 3 RY 2γ dx dr + β 2Y = 0 2γ dy dr (β 2 + 4β 4 )X = 2 + 2β 3 R I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

27 Straight-grad PT dimer: Solution in the Stokes variables Divide by Ṙ = 2γZ : Ẋ = β 2 YZ Ẏ = 2Z + (β 2 + 4β 4 )XZ + 2β 3 Z R Ż = 2Y + 2γR 4β 4 XY 2β 3 RY 2γ dx dr + β 2Y = 0 2γ dy dr (β 2 + 4β 4 )X = 2 + 2β 3 R Assuming ω 2 β 2 (β 2 + 4β 4 ) > 0, the general solution is ( ) ( ) ω ω X = A cos 2γ R + B sin 2γ R + 2 2β 3R β 2 + 4β 4 Y = ω ( ) ( )] ω ω [A sin β 2 2γ R B cos 2γ R + 4γβ 3 ω 2 Z = ± R 2 X 2 (R) Y 2 (R) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

28 Straight-grad PT dimer: integrals of motion X + iy = (A ib) exp (i ω2γ ) R + 2 2β 3R + i 4γβ 3 β 2 + 4β 4 ω 2 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

29 Straight-grad PT dimer: integrals of motion X + iy = (A ib) exp (i ω2γ ) R + 2 2β 3R + i 4γβ 3 β 2 + 4β 4 ω 2 Call (A ib) exp (i ω2γ ) R = ρe iθ ; then A ib = ρ exp ( iθ i ω2γ ) R I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

30 Straight-grad PT dimer: integrals of motion X + iy = (A ib) exp (i ω2γ ) R + 2 2β 3R + i 4γβ 3 β 2 + 4β 4 ω 2 Call (A ib) exp (i ω2γ ) R = ρe iθ ; then A ib = ρ exp ( iθ i ω2γ ) R The modulus and argument of A ib are integrals of motion: ( ρ 2 = X + 2β ) 3R 2 2 ( ) 2 ( β2 + Y 4γβ ) 2 3 β 2 + 4β 4 ω ω 2 H = 2γθ ωr I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

31 Straight-grad PT dimer: integrals of motion X + iy = (A ib) exp (i ω2γ ) R + 2 2β 3R + i 4γβ 3 β 2 + 4β 4 ω 2 Call (A ib) exp (i ω2γ ) R = ρe iθ ; then A ib = ρ exp ( iθ i ω2γ ) R The modulus and argument of A ib are integrals of motion: ( ρ 2 = X + 2β ) 3R 2 2 ( ) 2 ( β2 + Y 4γβ ) 2 3 β 2 + 4β 4 ω ω 2 H = 2γθ ωr ρ will be employed as a canonical variable and H as the Hamiltonian I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

32 Straight-grad PT dimer: integrals of motion X + iy = (A ib) exp (i ω2γ ) R + 2 2β 3R + i 4γβ 3 β 2 + 4β 4 ω 2 Call (A ib) exp (i ω2γ ) R = ρe iθ ; then A ib = ρ exp ( iθ i ω2γ ) R The modulus and argument of A ib are integrals of motion: ( ρ 2 = X + 2β ) 3R 2 2 ( ) 2 ( β2 + Y 4γβ ) 2 3 β 2 + 4β 4 ω ω 2 H = 2γθ ωr ρ will be employed as a canonical variable and H as the Hamiltonian (For standard dimer, H was identified in Ramezani, Kottos, El-Ganainy & Christodoulides, 2010 Phys Rev A ) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

33 Straight-grad PT dimer: Hamiltonian formulation Canonical coordinates: ρ, θ, P ρ, P θ I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

34 Straight-grad PT dimer: Hamiltonian formulation Canonical coordinates: Require Then ρ, θ, P ρ, P θ R = R(ρ, θ, P θ ) H = H(ρ, θ, P θ ). Ṗ ρ = H ρ decouples; ρ = H P ρ = 0 trivially satisfied I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

35 Straight-grad PT dimer: Hamiltonian formulation Canonical coordinates: Require Then ρ, θ, P ρ, P θ R = R(ρ, θ, P θ ) H = H(ρ, θ, P θ ). Ṗ ρ = H ρ decouples; ρ = H P ρ = 0 trivially satisfied Choose P θ = ψ such that θ = H ψ reproduces θ = ωz. The two equations equivalent if R ψ = Z I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

36 Straight-grad PT dimer: Hamiltonian formulation Canonical coordinates: Require Then ρ, θ, P ρ, P θ R = R(ρ, θ, P θ ) H = H(ρ, θ, P θ ). Ṗ ρ = H ρ decouples; ρ = H P ρ = 0 trivially satisfied Choose P θ = ψ such that θ = H ψ reproduces θ = ωz. The two equations equivalent if R ψ = Z Once θ = H ψ is satisfied, ψ = H θ follows automatically I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

37 Solving R/ ψ = Z Y = 4γβ 3 ω 2 + ω β 2 ρ cos θ, X = x 2β 3 β 2 + 4β 4 R, x = 2 β 2 + 4β 4 + ρ sin θ. I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

38 Solving R/ ψ = Z Y = 4γβ 3 ω 2 + ω β 2 ρ cos θ, X = x 2β 3 β 2 + 4β 4 R, x = 2 β 2 + 4β 4 + ρ sin θ. Substitute in X 2 + Y 2 + Z 2 = R 2, get constraint: ( λr + σ ) ( ) 2 R 2 λ x = r 2, ψ where r is independent of ψ: r 2 = x 2 /λ 2 + Y 2. I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

39 Solving R/ ψ = Z Y = 4γβ 3 ω 2 + ω β 2 ρ cos θ, X = x 2β 3 β 2 + 4β 4 R, x = 2 β 2 + 4β 4 + ρ sin θ. Substitute in X 2 + Y 2 + Z 2 = R 2, get constraint: ( λr + σ ) ( ) 2 R 2 λ x = r 2, ψ where r is independent of ψ: r 2 = x 2 /λ 2 + Y 2. Resolve the constraint: Z = r sinh(λψ), R = r λ cosh(λψ) σ λ 2 x. I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

40 Hamiltonian formulation of straight-grad dimer: summary Hamilton function: H(ρ, θ, P θ ) = 2γθ + ωσ λ 2 ( ) 2 + ρ sin θ β 2 + 4β 4 ω λ (4γβ3 ω 2 + ω ) 2 ρ cos θ + 1 ( ) 2 2 β 2 λ 2 + ρ sin θ cosh(λp θ ) β 2 + 4β 4 Hamilton equations: ρ = H P ρ, θ = H P θ, Ṗ ρ = H ρ, Ṗ θ = H θ. I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

41 Case study: Hamiltonian formulation of the standard dimer Standard dimer: i u + v + u 2 u = iγu i v + u + v 2 v = iγv I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

42 Case study: Hamiltonian formulation of the standard dimer Standard dimer: Hamilton function: i u + v + u 2 u = iγu i v + u + v 2 v = iγv H(ρ, θ, P θ ) = γθ ρ ρ sin θ cosh P θ Hamilton equations: ρ = H P ρ, θ = H P θ, Ṗ ρ = H ρ, Ṗ θ = H θ. (IB 2014 Phys Rev A ) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

43 Symmetry breaking: linear vs nonlinear Linear regime: Solutions: u, v e iωt, with ω 2 = 1 γ 2 iu t + v = iγu iv t + u = iγv I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

44 Symmetry breaking: linear vs nonlinear Linear regime: Solutions: u, v e iωt, with ω 2 = 1 γ 2 iu t + v = iγu iv t + u = iγv Symmetry-broken phase: γ > 1; symmetry-unbroken: γ < 1 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

45 Symmetry breaking: linear vs nonlinear Linear regime: Solutions: u, v e iωt, with ω 2 = 1 γ 2 iu t + v = iγu iv t + u = iγv Symmetry-broken phase: γ > 1; symmetry-unbroken: γ < 1 Nonlinear regime: standard dimer iu t + v + 2 u 2 u = iγu iv t + u + 2 v 2 v = iγv I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

46 Symmetry breaking: linear vs nonlinear Linear regime: Solutions: u, v e iωt, with ω 2 = 1 γ 2 iu t + v = iγu iv t + u = iγv Symmetry-broken phase: γ > 1; symmetry-unbroken: γ < 1 Nonlinear regime: standard dimer iu t + v + 2 u 2 u = iγu iv t + u + 2 v 2 v = iγv When γ > 1, all initial conditions blow up When γ < 1, small ICs remain bounded; large ICs blow up, e.g. I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

47 Symmetry breaking: linear vs nonlinear Linear regime: Solutions: u, v e iωt, with ω 2 = 1 γ 2 iu t + v = iγu iv t + u = iγv Symmetry-broken phase: γ > 1; symmetry-unbroken: γ < 1 Nonlinear regime: standard dimer iu t + v + 2 u 2 u = iγu iv t + u + 2 v 2 v = iγv When γ > 1, all initial conditions blow up When γ < 1, small ICs remain bounded; large ICs blow up, e.g. { } u(t) = exp { γ(t t 0 ) + i γ sinh[2γ(t t 0)] } v(t) = exp γ(t t 0 ) + i γ sinh[2γ(t t 0)] (IB, G Jackson & S Flach 2013 Phys Rev A ) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

48 Nonlinear softening of PT -symmetry breaking transition i u + v + ( u v 2 )u + v 2 u = iγu, i v + u + ( v u 2 )v + u 2 v = iγv. I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

49 Nonlinear softening of PT -symmetry breaking transition i u + v + ( u v 2 )u + v 2 u = iγu, i v + u + ( v u 2 )v + u 2 v = iγv. Stokes variable: Ẍ + ν 2 X = 0, ν 2 = (2 + Z ) 2 4γ 2 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

50 Nonlinear softening of PT -symmetry breaking transition i u + v + ( u v 2 )u + v 2 u = iγu, i v + u + ( v u 2 )v + u 2 v = iγv. Stokes variable: Ẍ + ν 2 X = 0, ν 2 = (2 + Z ) 2 4γ 2 (2 + Z ) 2 > 4γ 2 : X = ρ 0 cos ντ, Y = Y Z ρ 0 sin ντ ν 2(γ+1) < Z < 2(γ 1) : X = A sinh στ, Y = 2 + Z σ A cosh στ +Y 0 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

51 Nonlinear PT -symmetry breaking Ball X 2 (0) + Y 2 (0) + Z 2 (0) r 2 : r < 2(γ 1) small; r 2(γ 1) large; r c = 2(γ 1) critical I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

52 Nonlinear PT -symmetry breaking Ball X 2 (0) + Y 2 (0) + Z 2 (0) r 2 : r < 2(γ 1) small; r 2(γ 1) large; r c = 2(γ 1) critical Nonlinear PT -symmetry breaking: γ c (P) = 1 + P I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

53 Nonlinearity-induced PT -symmetry restoration A 4-parameter family of cross-grad dimers: i u + v iγu = α 3 ( u v 2 )u + α 3 v 2 u [ +2α 4 u 2 v + (2α 1 + α 2 + 2α 4 ) u 2 + 2α 1 v 2] v i v + u + iγv = α 3 (2 u 2 + v 2 )v + α 3 u 2 v [ +2α 4 v 2 u + (2α 1 + α 2 + 2α 4 ) v 2 + 2α 1 u 2] u I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

54 Nonlinearity-induced PT -symmetry restoration A 4-parameter family of cross-grad dimers: i u + v iγu = α 3 ( u v 2 )u + α 3 v 2 u [ +2α 4 u 2 v + (2α 1 + α 2 + 2α 4 ) u 2 + 2α 1 v 2] v i v + u + iγv = α 3 (2 u 2 + v 2 )v + α 3 u 2 v [ +2α 4 v 2 u + (2α 1 + α 2 + 2α 4 ) v 2 + 2α 1 u 2] u Hamiltonian expressed in the Stokes variables: ( H = α 1 R 2 α2 ) α 4 Z 2 + α 3 Z R + α 2 4 Y 2 R γy I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

55 Nonlinearity-induced PT -symmetry restoration A 4-parameter family of cross-grad dimers: i u + v iγu = α 3 ( u v 2 )u + α 3 v 2 u [ +2α 4 u 2 v + (2α 1 + α 2 + 2α 4 ) u 2 + 2α 1 v 2] v i v + u + iγv = α 3 (2 u 2 + v 2 )v + α 3 u 2 v [ +2α 4 v 2 u + (2α 1 + α 2 + 2α 4 ) v 2 + 2α 1 u 2] u Hamiltonian expressed in the Stokes variables: ( H = α 1 R 2 α2 ) α 4 Z 2 + α 3 Z R + α 2 4 Y 2 R γy Lower bound: ( H α 1 α ) [ 2 R + α 3Z γ 1 4 2(α α 2) ] 2 ( + α 4 + α 2 4 ) Z 2 (α 3Z γ 1) 2 α 2 + 4α 1 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

56 Nonlinearity-induced PT -symmetry restoration A 4-parameter family of cross-grad dimers: i u + v iγu = α 3 ( u v 2 )u + α 3 v 2 u [ +2α 4 u 2 v + (2α 1 + α 2 + 2α 4 ) u 2 + 2α 1 v 2] v i v + u + iγv = α 3 (2 u 2 + v 2 )v + α 3 u 2 v [ +2α 4 v 2 u + (2α 1 + α 2 + 2α 4 ) v 2 + 2α 1 u 2] u The PT -symmetry is restored nonlinearly if: α 1 and α 2 are both nonzero and have the same sign α 1 and α 2 both nonzero and of the opposite sign, with α 2 < 4 α 1 α 1 0 while α 2 = 0 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

57 PT -symmetry restoration: straight-grad dimers i u + v iγu = i v + u + iγv = [ 2β 1 u 2 + (2β 1 + β 2 + 2β 4 ) v 2] u +2β 4 v 2 u + β 3 u 2 v + β 3 (2 u 2 + v 2) v [ 2β 1 v 2 + (2β 1 + β 2 + 2β 4 ) u 2] v +2β 4 u 2 v + β 3 v 2 u + β 3 (2 v 2 + u 2) u I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

58 PT -symmetry restoration: straight-grad dimers i u + v iγu = i v + u + iγv = [ 2β 1 u 2 + (2β 1 + β 2 + 2β 4 ) v 2] u +2β 4 v 2 u + β 3 u 2 v + β 3 (2 u 2 + v 2) v [ 2β 1 v 2 + (2β 1 + β 2 + 2β 4 ) u 2] v +2β 4 u 2 v + β 3 v 2 u + β 3 (2 v 2 + u 2) u Parametric trajectory (R 2 = X 2 + Y 2 + Z 2 ): ( ) ( ) ω ω X = A cos 2γ R + B sin 2γ R + 2 2β 3R, β 2 + 4β 4 Y = ω ( ) ( )] ω ω [A sin β 2 2γ R B cos 2γ R + 4γβ 3 ω 2, I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

59 PT -symmetry restoration: straight-grad dimers i u + v iγu = i v + u + iγv = [ 2β 1 u 2 + (2β 1 + β 2 + 2β 4 ) v 2] u +2β 4 v 2 u + β 3 u 2 v + β 3 (2 u 2 + v 2) v [ 2β 1 v 2 + (2β 1 + β 2 + 2β 4 ) u 2] v +2β 4 u 2 v + β 3 v 2 u + β 3 (2 v 2 + u 2) u Parametric trajectory (R 2 = X 2 + Y 2 + Z 2 ): ( ) ( ) ω ω X = A cos 2γ R + B sin 2γ R + 2 2β 3R, β 2 + 4β 4 Y = ω ( ) ( )] ω ω [A sin β 2 2γ R B cos 2γ R + 4γβ 3 ω 2, X = 2β 3 β 2 + 4β 4 R + O(R 0 ) as R, 2β 3 impossible if > 1 β 2 + 4β 4 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

60 PT -symmetry restoration: straight-grad dimers i u + v iγu = i v + u + iγv = [ 2β 1 u 2 + (2β 1 + β 2 + 2β 4 ) v 2] u +2β 4 v 2 u + β 3 u 2 v + β 3 (2 u 2 + v 2) v [ 2β 1 v 2 + (2β 1 + β 2 + 2β 4 ) u 2] v +2β 4 u 2 v + β 3 v 2 u + β 3 (2 v 2 + u 2) u The PT -symmetry is restored nonlinearly if: β 2 (β 2 + 4β 3 ) 0 and 2 β 3 > β 2 + 4β 4 β 2 = 0 but β 3 0 β 2 + 4β 4 = 0 but β 2 0 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

61 PT -symmetry restoration: straight-grad dimers β 2 (β 2 + 4β 4 ) > 0 2β 3 β+2+4β 4 < 1 2β 3 β+2+4β 4 > 1 2β 3 β+2+4β 4 = 1 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

62 PT -symmetry restoration: straight-grad dimers β 2 (β 2 + 4β 4 ) > 0 2β 3 β+2+4β 4 < 1 2β 3 β+2+4β 4 > 1 2β 3 β+2+4β 4 = 1 β 2 (β 2 + 4β 4 ) < 0 I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

63 Summary of results / Conclusions A 4-parameter family of cross-grad and a 4-parameter family of straight-grad dimers shown to be Hamiltonian completely integrable systems I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

64 Summary of results / Conclusions A 4-parameter family of cross-grad and a 4-parameter family of straight-grad dimers shown to be Hamiltonian completely integrable systems Nonlinearity may soften the PT -symmetry breaking transition: stable periodic and quasiperiodic states persist for an arbitrarily large value of the gain-loss coefficient I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

65 Summary of results / Conclusions A 4-parameter family of cross-grad and a 4-parameter family of straight-grad dimers shown to be Hamiltonian completely integrable systems Nonlinearity may soften the PT -symmetry breaking transition: stable periodic and quasiperiodic states persist for an arbitrarily large value of the gain-loss coefficient Nonlinearity may suppress the PT -symmetry breaking completely. The PT -symmetry becomes spontaneously restored I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21

66 Summary of results / Conclusions A 4-parameter family of cross-grad and a 4-parameter family of straight-grad dimers shown to be Hamiltonian completely integrable systems Nonlinearity may soften the PT -symmetry breaking transition: stable periodic and quasiperiodic states persist for an arbitrarily large value of the gain-loss coefficient Nonlinearity may suppress the PT -symmetry breaking completely. The PT -symmetry becomes spontaneously restored Based on: IB, PRA (2014) IB, D E Pelinovsky and P Dubard, submitted IB and M Gianfreda, Journ Phys A (2014) I Barashenkov (UCT) Integrability & PT -symmetry restoration Palermo / 21