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

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

Introduction Numerical Methods Numerical Simulations/Results Future Work Acknowledgment Outline Introduction 1 CCQGLE Classes of Solitons Solutions No Hopf Bifurcations in Hamiltonian Systems Numerical Methods 2 Simulations on 2D CCQGLE Initial Conditions Parameters Numerical Simulations/Results 3 2D Solitons Future Work 4 3D CCQGLE Acknowledgment 5 Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods CCQGLE Numerical Simulations/Results Classes of Solitons Solutions Future Work No Hopf Bifurcations in Hamiltonian Systems Acknowledgment Complex Cubic-Quintic Ginzburg-Landau Equation (CCQGLE) CCQGLE ∂ t A = ǫ A + ( b 1 + ic 1 ) ∇ 2 ⊥ A − ( b 3 − ic 3 ) | A | 2 A − ( b 5 − ic 5 ) | A | 4 A 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, b 1 − angular spectral filtering, c 1 = 0 . 5 − diffraction coefficient, b 3 − nonlinear gain/loss, c 3 = 1 − nonlinear dispersion, b 5 − saturation of the nonlinear gain/loss, c 5 − saturation of the nonlinear refractive index Akhmediev et. al. [1] new classes: pulsating, creeping, snaking, chaotical Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods CCQGLE Numerical Simulations/Results Classes of Solitons Solutions Future Work No Hopf Bifurcations in Hamiltonian Systems Acknowledgment Previous Numerical Simulations on 1D CCQGLE Figure: Pulsating, Snaking, Creeping, Chaotical Mancas, ERAU 2012 2D Solitons

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

Introduction Numerical Methods Simulations on 2D CCQGLE Numerical Simulations/Results Initial Conditions Future Work Parameters Acknowledgment 2D Fourier Spectral Method Fourier � ∞ � ∞ 1 −∞ e − i ( k x x + k y y ) u ( x , y ) dxdy F ( u )( k x , k y ) = � u ( k x , k y ) = 2 π −∞ inverse Fourier � ∞ � ∞ 1 F − 1 ( � −∞ e i ( k x x + k y y ) � u )( x , y ) = u ( x , y ) = u ( k x , k y ) dk x dk y 2 π −∞ PDE ⇒ ODE A + β � | A | 2 A + γ � A t = α ( k x , k y ) � � | A | 4 A α ( k x , k y ) = ǫ − ( b 1 + ic 1 )( k 2 x + k 2 y ) , β = − ( b 3 − ic 3 ) , γ = − ( b 5 − ic 5 ) Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Simulations on 2D CCQGLE Numerical Simulations/Results Initial Conditions Future Work Parameters Acknowledgment Spatial Discretization (Discrete Fourier Transform) Rectangular Mesh Ω = [ − L / 2 , L / 2 ] × [ − L / 2 , L / 2 ] into n × n uniformly spaced grid points X ij = ( x i , y j ) with ∆ x = ∆ y = L / n , and A ( X ij ) = A ij 2DFT A k x k y = ∆ x ∆ y � n � n � j = 1 e − i ( k x x i + k y y j ) A ij , k x , k y = − n 2 + 1 , · · · , n i = 1 2 inverse 2DFT � n / 2 � n / 2 k y = − n / 2 + 1 e i ( k x x i + k y y j ) � 1 A ij = A k x k y , i , j = ( 2 π ) 2 k x = − n / 2 + 1 1 , 2 , · · · , n Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Simulations on 2D CCQGLE Numerical Simulations/Results Initial Conditions Future Work Parameters Acknowledgment Temporal Discretization Explicit scheme for the nonlinear part, and exact solution for the � linear part � A ( x , y ; 0 ) e α ( k x , k y ) t A ( t ) = A n = � Initializing � A ( t n ) ⇒ �� � �� � � � 2 4 � � � � � F − 1 ( � F − 1 ( � � F − 1 ( � F − 1 ( � A n ) A n ) A n ) A n ) N 3 = F , N 5 = F � � 4 step AB, or 4th order RK A n + 1 = � � � A n e α ( k x , k y ) t + ∆ t � 55 f ( � A n ) − 59 f ( � A n − 1 ) + 37 f ( � A n − 2 ) − 9 f ( � A n − 3 ) 24 f ( � A ) = β N 1 + γ N 2 Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Simulations on 2D CCQGLE Numerical Simulations/Results Initial Conditions Future Work Parameters Acknowledgment IC Gaussian A ( x , y ; 0 ) = A 0 e − r 2 ring shape with rotating phase A ( x , y ; 0 ) = A 0 r m e − r 2 e im θ m − degree of vorticity, A 0 − real amplitude, θ = tan − 1 � σ y y � σ x x widths either circular or elliptic are controlled by � ( σ x x ) 2 + ( σ y y ) 2 r = Figure: Initial shapes of solitons. Left: Gaussian, Right: Ring with vorticity m = 1. Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Simulations on 2D CCQGLE Numerical Simulations/Results Initial Conditions Future Work Parameters Acknowledgment System’s Parameters Initial parameters Monitor energy � ∞ � ∞ −∞ | A ( x , y ; t ) | 2 dxdy = � n � n j = 1 | A ij | 2 ∆ x ∆ y Q ( t ) = i = 1 −∞ 2D solitons b 1 c 1 b 3 c 3 b 5 c 5 ǫ 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

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Stationary Solitons circular Gaussian IC and stays radially symmetric, stable and uninteresting. A 0 = 2 . 5, and σ x = σ y = 1 Figure: Energy is concentrated in the center of the domain Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Ring Vortex (stable) Solitons Circular ring with rotating phase IC but different parameters, stable, it is spinning around its center. A 0 = 2 . 5, and σ x = σ y = 1 Figure: Top Left: Energy. Top Right: Ring vortex at t = 20 s . Bottom Left: Contour plot of | A | 2 . Bottom Right: Phase plot of θ at t = 20 s Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Ring Vortex (unstable) Solitons Circular Vortex it is spinning so much that breaks its symmetry changes into several bell-shaped solitons via multiple bifurcations, A 0 = 3, σ x = 0 . 15, σ y = 0 . 15 Figure: Left: 10 bell-shaped solitons due to defocusing. Right: phases are not spinning Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Ring Vortex (stable) Solitons Elliptic stable, it is spinning around its center, and breaks symmetry but remains stable, A 0 = 2 . 5, σ x = 0 . 15, σ y = 0 . 85 Figure: Two peaks appear on top Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Pulsating Solitons (change stability) Gaussian IC, A 0 = 5, slightly elliptical, σ x = 0 . 8333 and σ y = 0 . 9091 Pulsating similar to stationary initially but requires longer time to capture pulsations Figure: Left: Energy shows transitions. Right: No pulsations at t = 200 s Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Pulsating Phase at t = 480 s , t = 490 s Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Pulsating Phase at t = 500 s , t = 510 s Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Parameters for Exploding/Erupting Gaussian IC, A 0 = 3 . 0, and circular σ x = σ y = 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

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Energy for Exploding/Erupting Gaussian IC, A 0 = 3 . 0, and circular σ x = σ y = 0 . 3 Exploding: look for high bursts of energy Figure: Energy is periodic with high bursts almost every 12 s Mancas, ERAU 2012 2D Solitons

Introduction Numerical Methods Numerical Simulations/Results 2D Solitons Future Work Acknowledgment Exploding/Erupting Initial soliton is smooth, then circular waves appear and grow. Figure: Evolution for the exploding t = 90 s , t = 91 s Mancas, ERAU 2012 2D Solitons