2D Solitons in Dissipative Media Stefan C. Mancas Nonlinear Waves - - PowerPoint PPT Presentation

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2d solitons in dissipative media
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2D Solitons in Dissipative Media Stefan C. Mancas Nonlinear Waves - - PowerPoint PPT Presentation

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons in Dissipative Media Stefan C. Mancas Nonlinear Waves Department of Mathematics Embry-Riddle Aeronautical University Daytona Beach, FL.

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment

2D Solitons in Dissipative Media

Stefan C. Mancas

Nonlinear Waves Department of Mathematics Embry-Riddle Aeronautical University Daytona Beach, FL. 32114

XIV th International Conference Geometry, Integrability and Quantization June 8-13, 2012 Varna, Bulgaria

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment

Outline

1

Introduction CCQGLE Classes of Solitons Solutions No Hopf Bifurcations in Hamiltonian Systems

2

Numerical Methods Simulations on 2D CCQGLE Initial Conditions Parameters

3

Numerical Simulations/Results 2D Solitons

4

Future Work 3D CCQGLE

5

Acknowledgment

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment CCQGLE Classes of Solitons Solutions No Hopf Bifurcations in Hamiltonian Systems

Complex Cubic-Quintic Ginzburg-Landau Equation (CCQGLE)

CCQGLE ∂tA = ǫA + (b1 + ic1)∇2

⊥A − (b3 − ic3)|A|2A − (b5 − ic5)|A|4A

Canonical equation governing the weakly nonlinear behavior of dissipative systems ∇2

⊥−transverse Laplacian for radially symmetric beams,

A(x, y; t)−envelope field, t−cavity number ǫ−linear loss/gain, b1−angular spectral filtering, c1 = 0.5− diffraction coefficient, b3−nonlinear gain/loss, c3 = 1−nonlinear dispersion, b5−saturation of the nonlinear gain/loss, c5−saturation of the nonlinear refractive index Akhmediev et. al. [1] new classes: pulsating, creeping, snaking, chaotical

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment CCQGLE Classes of Solitons Solutions No Hopf Bifurcations in Hamiltonian Systems

Previous Numerical Simulations on 1D CCQGLE

Figure: Pulsating, Snaking, Creeping, Chaotical

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment CCQGLE Classes of Solitons Solutions No Hopf Bifurcations in Hamiltonian Systems

Hamiltonian Systems → No Hopf Bifurcations

Five classes of solutions that are not stationary in time Don’t exist as stable structures in Hamiltonian systems Envelopes exhibit complicated temporal dynamics and are unique to dissipative systems Dissipation allows the occurrence of Hopf and it leads to the various classes of pulsating solitons in CCQGLE

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment Simulations on 2D CCQGLE Initial Conditions Parameters

2D Fourier Spectral Method

Fourier F(u)(kx, ky) = u(kx, ky) =

1 2π

∞

−∞

∞

−∞ e−i(kxx+kyy)u(x, y) dxdy

inverse Fourier F−1( u)(x, y) = u(x, y) =

1 2π

∞

−∞

∞

−∞ ei(kxx+kyy)

u(kx, ky) dkxdky PDE ⇒ ODE

  • At = α(kx, ky)

A + β |A|2A + γ |A|4A α(kx, ky) = ǫ − (b1 + ic1)(k2

x + k2 y ), β = −(b3 − ic3),

γ = −(b5 − ic5)

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment Simulations on 2D CCQGLE Initial Conditions Parameters

Spatial Discretization (Discrete Fourier Transform)

Rectangular Mesh Ω = [−L/2, L/2] × [−L/2, L/2] into n × n uniformly spaced grid points Xij = (xi, yj) with ∆x = ∆y = L/n, and A(Xij) = Aij 2DFT

  • Akxky = ∆x∆y n

i=1

n

j=1 e−i(kxxi+kyyj)Aij, kx, ky = − n 2 + 1, · · · , n 2

inverse 2DFT Aij =

1 (2π)2

n/2

kx=−n/2+1

n/2

ky=−n/2+1 ei(kxxi+kyyj)

Akxky, i, j = 1, 2, · · · , n

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment Simulations on 2D CCQGLE Initial Conditions Parameters

Temporal Discretization

Explicit scheme for the nonlinear part, and exact solution for the linear part A(t) =

  • A(x, y; 0)eα(kx,ky)t

Initializing An = A(tn) ⇒ N3 = F

  • F−1(

An)

  • 2

F−1( An)

  • , N5 = F
  • F−1(

An)

  • 4

F−1( An)

  • 4 step AB, or 4th order RK
  • An+1 =
  • Aneα(kx,ky)t + ∆t

24

  • 55f(

An) − 59f( An−1) + 37f( An−2) − 9f( An−3)

  • f(

A) = βN1 + γN2

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment Simulations on 2D CCQGLE Initial Conditions Parameters

IC

Gaussian A(x, y; 0) = A0e−r 2 ring shape with rotating phase A(x, y; 0) = A0r me−r 2eimθ m− degree of vorticity, A0− real amplitude, θ = tan−1 σyy

σxx

  • widths either circular or elliptic are controlled by

r =

  • (σxx)2 + (σyy)2

Figure: Initial shapes of solitons. Left: Gaussian, Right: Ring with

vorticity m = 1.

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment Simulations on 2D CCQGLE Initial Conditions Parameters

System’s Parameters

Initial parameters Monitor energy Q(t) = ∞

−∞

∞

−∞ |A(x, y; t)|2 dxdy = n i=1

n

j=1 |Aij|2∆x∆y

2D solitons ǫ b1 c1 b3 c3 b5 c5 Stationary

  • 0.045

0.04 0.5

  • 0.21

1 0.03

  • 0.08

Vortex (spinning)

  • 0.1

0.1 0.5

  • 0.88

1 0.04

  • 0.02

Pulsating

  • 0.045

0.04 0.5

  • 0.37

1 0.05

  • 0.08

Exploding/Erupting

  • 0.1

0.125 0.5

  • 1

1 0.1

  • 0.6

Creeping

  • 0.1

0.101 0.5

  • 1.3

1 0.3

  • 0.101

Table: Initial sets of parameters for 2D solitons from which we start

simulations [1]

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Stationary Solitons

circular Gaussian IC and stays radially symmetric, stable and

  • uninteresting. A0 = 2.5, and σx = σy = 1

Figure: Energy is concentrated in the center of the domain

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Ring Vortex (stable) Solitons

Circular ring with rotating phase IC but different parameters, stable, it is spinning around its center. A0 = 2.5, and σx = σy = 1

Figure: Top Left: Energy. Top Right: Ring vortex at t = 20s. Bottom

Left: Contour plot of |A|2. Bottom Right: Phase plot of θ at t = 20s

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Ring Vortex (unstable) Solitons

Circular Vortex it is spinning so much that breaks its symmetry changes into several bell-shaped solitons via multiple bifurcations, A0 = 3, σx = 0.15, σy = 0.15

Figure: Left: 10 bell-shaped solitons due to defocusing. Right: phases

are not spinning

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Ring Vortex (stable) Solitons

Elliptic stable, it is spinning around its center, and breaks symmetry but remains stable, A0 = 2.5, σx = 0.15, σy = 0.85

Figure: Two peaks appear on top

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Pulsating Solitons (change stability)

Gaussian IC, A0 = 5, slightly elliptical, σx = 0.8333 and σy = 0.9091 Pulsating similar to stationary initially but requires longer time to capture pulsations

Figure: Left: Energy shows transitions. Right: No pulsations at t = 200s

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Pulsating Phase at t = 480s, t = 490s

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Pulsating Phase at t = 500s, t = 510s

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Parameters for Exploding/Erupting

Gaussian IC, A0 = 3.0, and circular σx = σy = 0.3 computed over 64 simulations within a 5 dimensional space by varying parameters one by one and looked for right Q(t)

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Energy for Exploding/Erupting

Gaussian IC, A0 = 3.0, and circular σx = σy = 0.3 Exploding: look for high bursts of energy

Figure: Energy is periodic with high bursts almost every 12s

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Exploding/Erupting

Initial soliton is smooth, then circular waves appear and grow.

Figure: Evolution for the exploding t = 90s, t = 91s

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Exploding/Erupting

Envelopes begin to degenerate, going from a radially Gaussian shape to regions of its slopes that cave in

Figure: Evolution for the exploding t = 92s, t = 93s

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Exploding/Erupting

Then, soliton explodes intermittently, resulting in significant bursts of power above, but it recovers the initial shape after the explosion

Figure: Evolution for the exploding t = 94s, t = 95s

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Exploding/Erupting

a) b) c) d) e) f)

Figure: Evolution for the exploding soliton

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Creeping

Gaussian IC, A0 = 3.0, and circular σx = σy = 0.25 Creeping Soliton for 0-100 s Creeping Soliton for 100-200 s It changes its shape and shifts a finite distance periodically while remains confined to domain

Figure: Creeping soliton

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 2D Solitons

Energy for Creeping

Figure: Energy for creeping soliton

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment 3D CCQGLE

3D Solitons

Vary parameters for all classes of solitons Increase vorticity m > 1 Study the stability regimes, transitions to instability, breaking, emerging a new class or non-existing (dissipating) Develop 3D numerical schemes, light bullets Soliton-soliton interaction

Mancas, ERAU 2012 2D Solitons

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Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment

Acknowledgment

Computations were performed on a Linux cluster (256 Intel Xeon 3.2GHz 1024 KB cache 4GB with Myrinet MX, GNU Linux) at ERAU

Figure: ZEUS cluster at ERAU

Work was partially supported by Office of Sponsored Research, ERAU

Mancas, ERAU 2012 2D Solitons

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Appendix References

References I

J.M. Soto-Crespo, N. Akhmediev, A. Ankiewicz Pulsating, creeping, and erupting solitons in dissipative systems

  • JPhys. Rev. Lett., 85:2937, 2000.

J.M. Soto-Crespo, N. Akhmediev , N. Devine, Mejia-Cortis Transformations of continuously self-focusing and continuously self-defocusing dissipative solitons Optics Express, 16:15388, 2008.

Mancas, ERAU 2012 2D Solitons