Dissipative Solitons in Physical Systems

Dissipative Solitons in Physical Systems Talk given by H.-G. Purwins Institute of Applied Physics University of Münster, Germany

Chapter 1 Introduction

Cosmology and Pattern Formation 1-1

Complex Behaviour of Pattern Forming Nonlinear Dissipative Systems Patterns in space and time Attractors Complex dependence on initial conditions boundary conditions parameters Dependence on initial conditions multistability Dependence on parameters and boundary conditions bifurcation Lack of reproducibility in the presence of noise Chaos Understanding of self-organized patterns is one of the most important problems of modern science 1-2

Chapter 2 Gas-Discharge Systems

Lichtenberg Pattern I: Experimental Set-Up elektrophorus Lichtenberg (1742-1749) 2-1

Lichtenberg (1742-1749) 2-2 Lichtenberg Pattern II: Reproduction of the Original Pattern

Current Filaments in a Pulse Driven Planar Gas-Discharge System with Dielectric Barrier photograph of the luminescence radiation from the discharge space measured on one electrode Buss 1932 2-3

Chapter 3 Experimental Results from Gas-Discharge Systems with High Ohmic Barrier

Experimental Set-Up for Measuring Self-Organized Patterns in Planar DC Gas-Discharge Systems 3-1

Superposition of 1000 Subsequent Experimental Frames of a Single Propagating Current Filament coordinate y / mm monotonous tails coordinate y / mm oscillatory tails current density j / Am -2 coordinate x / mm parameters: U 0 = 2,74 kv, ρ SC = 4,95 MΩ cm, R 0 = 20 MΩ, Gas: N 2, T=100 K, p = 280 hpa, D=30 mm, d = 250 µm, a SC =1 mm, I = 46 µa, t exp =20 ms current density j / Am -2 coordinate x / mm parameters: U 0 = 3,6 kv,ρ SC = 3,05 MΩ cm, R 0 = 4,4 MΩ, Gas: N 2, T=100 K, p = 279 hpa, D=30 mm, d = 500 µm, a SC =1 mm, I = 200 µa, t exp =20 ms Bödeker et al. 2003 3-2

Typical Dynamic Behaviour of a Single Experimental Current Filament in the Discharge Plane 1 cm real time parameters: U 0 =2,7 kv, ρ SC =4,95 MΩ cm, R 0 =20 MΩ, Gas: N 2, T=100 K, p=280 hpa, D=30 mm, d=250 µm, a SC =1 mm, I=46 µa Bödeker et al. 2003 3-3

Typical Scattering Process for two Interacting Filaments in the Discharge Plane t=0 ms t=120 ms t=200 ms t=248 ms t=300 ms t=360 ms parameters: U 0 =3 kv, ρ SC =1.98 MΩ cm, R 0 =4.4 MΩ, Gas: N 2, T=100 K, p=244 hpa, D=30 mm, d=500 µm, a SC =1 mm, I=140 µa, t exp =20 ms, f rep =50 Hz Bödeker, Purwins 2003 3-4

Typical Process for Molecule Formation of two Interacting Filaments in the Discharge Plane t=0 ms t=2 s t=5.68 s t=7.9 s t=9.82 s t=10.42 s parameters: U 0 =3.1 kv, ρ SC =4.19 MΩ cm, R 0 =4.4 MΩ, Gas: N 2, T=100 K, p=290 hpa, D=30 mm, d=500 µm, a SC =1 mm, I=170 µa, t exp =20 ms, f rep =50 Hz Bödeker, Purwins 2003 3-5

Typical Process for the Generation of a Current Filament in the Discharge Plane t=0 ms t=127.5 ms t=212.5 ms t=271 ms t=274 ms t=275 ms parameters: U 0 =3,8 kv, ρ SC =4,14 MΩ cm, R 0 =20 MΩ, Gas: N 2, T=100 K, p=290 hpa, D=30 mm, d=500 µm, a SC =1 mm, I=100-250 µa, t exp =0,2 ms, f rep =2 khz Bödeker, Purwins 2003 3-6

Typical Process for the Annihilation of a Current Filament in the Discharge Plane t=0 ms t=93 ms t=211.5 ms t=293 ms t=306 ms t=306.5 ms parameters: U 0 =3,8 kv, ρ SC =4,14 MΩ cm, R 0 =20 MΩ, Gas: N 2, T=100 K, p=290 hpa, D=30 mm, d=500 µm, a SC =1 mm, I=100-250 µa, t exp =0,2 ms, f rep =2 khz Bödeker, Purwins 2003 3-7

Drift Bifurcation of an Experimental Filament in the Discharge Plane Obtained from Stochastic Data Analysis of Trajectories 2 square of velocity v 0 / mm 2 s -2 parameters: U 0 =3,7 kv, R 0 =10 MΩ, Gas: N 2, T=100 K, p=286 hpa, D=30 mm, d=750 µm, a SC =1 mm, I=107 µa, t exp =20 ms, f rep =50 Hz specific resistivity ρ SC / MΩcm square of the intrinsic velocity as a function of the specific resistivity of the semiconductor wafer and typical experimental trajectories Bödeker et al. 2003 3-8

Strümpel, Purwins 2000 3-9 Experimentally Observed Hierarchy of Filamentary Patterns in the DC Gas-Discharge System I

Strümpel, Purwins 2000 3-10 Experimentally Observed Hierarchy of Filamentary Patterns in the DC Gas-Discharge System II

Strümpel, Purwins 2000 3-11 Experimentally Observed Hierarchy of Filamentary Patterns in the DC Gas-Discharge System III

Chapter 4 Qualitative Model for Planar Gas-Discharge Systems with High Ohmic Barrier

Purwins, Klempt, Berkemeier 1987 4-1 The Local DC Gas-Discharge System as Activator-Inhibitor System

The 3-Component Reaction-Diffusion-Equation (3-k-RD-System) κ2 ut = Du u + f(u) κ 3v κ4 w + κ 1 Ω Ω ud, τ v = D v + u v, θ w = D w+ u w, t t v w Ω 3 f(u) λ u u, u = u(x;t), v = v(x;t), w = w(x;t), x Ω, D,D,D, τ, θ, λ, κ, κ, κ 0. u v w 2 3 4 2 R R, R 1 3, Bode et al. 2002 4-2

Numerical Solution of the 3-k-RD-System: Dissipative Soliton as Localized Travelling Solitary Structure in R 3 τ=48.0, θ=0.5, D u =1.5*10-4, D v =1.86*10-4, D w =9.6*10-3, λ=2.0, κ 1 =-6.92, κ 2 =0, κ 3 =8.5, κ 4 =1.0, Ω=[0,0.466]x[0,0.932], x=0.0155, t=0.01. Liehr et al. 2003 4-3

Numerical Solutions of the 3-k-RD-System: Scattering of Interacting Dissipative Solitons in R 2 t=100 t=300 t=600 t=900 t=1200 t=1480 τ=3.35, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0 Ω=[0,1]x[0,1], x=5*10-3, t=0.1. Liehr et al. 2003 4-4

Numerical Solutions of the 3-k-RD-System: Formation of Rotating Molecules Due to Collision of Dissipative Solitons in R 2 t=100 t=320 t=540 t=760 t=980 t=1200 τ=3.35, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[0,1]x[0,1], x=5*10-3, t=0.1. Liehr et al. 2003 4-5

Numerical Solutions of the 3-k-RD-System: Generation of a Dissipative Soliton Due to Collision in R 2 t=105 t=250 t=265 t=285 t=405 t=530 τ=3.47, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[0,1]x[0,1], x=5*10-3, t=0.1. Liehr et al. 2003 4-6

Numerical Solutions of the 3-k-RD-System: Annihilation of a Dissipative Soliton Due to Collision in t=60 t=192 t=324 R 2 t=456 t=588 t=720 τ=3.59, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[0,1]x[0,1], x=5*10-3, t=0.1. Liehr 2003 4-7

Drift Bifurcation of an Experimental Filament in the Discharge Plane Obtained from Stochastic Data Analysis of Trajectories 2 square of velocity v 0 / mm 2 s -2 parameters: U 0 =3,7 kv, R 0 =10 MΩ, Gas: N 2, T=100 K, p=286 hpa, D=30 mm, d=750 µm, a SC =1 mm, I=107 µa, t exp =20 ms, f rep =50 Hz specific resistivity ρ SC / MΩcm square of the intrinsic velocity as a function of the specific resistivity of the semiconductor wafer and typical experimental trajectories Bödeker et al. 2003 4-8

Relevance of the 3-k-RD-System exemplaric theoretical investigation 3-component nonlinear partial differential equation simple structure solutions reflect a large variety of particle properties strong relation to electrical transport systems theoretical predictions could be manifested experimentally observed phenomena could be found in the solutions of the equations expectation: deep insight into the formation of self-organized patterns deep insight into the mechanisms of pattern formation of nonlinear dissipative systems in general 4-9

Chapter 5 The Reduced Equation

The 3-Component Reaction-Diffusion-Equation (3-k-RD-System) 2 ut = Du u + f(u) κ3v κ 4w + κ1 ud Ω, Ω Ω τ v = D v + u v, t t κ = + κ κ + κ Ω v θ w = D w + u w, w λ = = = Ω R R 3 f(u) u u, 1 2 3 u u(x;t), v v(x;t), w w(x;t), x,,, R D,D,D, τθλκ,,,, κ, κ 0. u v w 2 3 4 Bode et al. 2002 5-1

Reduction of the Field Equation to a Dynamical Equation for Dissipative Solitons I ansatz: normalization of the homogeneous stationary background to 0, 2 3 u(x,t 1,T 2,T 3) = u(x p 1) + u(x p 2) +ε ru ε R u, v(x,t,t,t ) = u(x p ) + u(x p ) +εα u(x p ) +εα u(x p ) +ε r +ε R. 2 3 1 2 3 1 2 1 1 2 2 v v time scale separation: j pi = p i(t 1,T 2,T 3), i = 1,2, ru,v = r u,v(x,t 1), Tj =ε t, j = 1,2,3, εα =εα (T,T ), R = R (x). i i 1 2 u,v u,v visualization of and in R for a dissipative Soliton Bode et al. 2002 α 2 p isoline of the activator u x 2 x 1 5-2 isoline of the inhibitor v

Reduction of the Field Equation to a Dynamical Equation for Dissipative Solitons II equation of motion for N dissipative solitons p =κ α W (p,...,p ), i = 1,2,...,N i 3 i i 1 N 2 1 2 α i = κ 3 τ α i κ 3Q α i α i W i(p 1,...,p N) κ3 interaction function W (p,...,p ) W (p,p ) F p p ( ) N N j i i 1 N = DS i j = j i j = 1 j = 1 pj pi j i j i p p Bode et al. 2002 5-3

Interaction Function of the Equation of Motion for Dissipative Solitons Derived from the Field Equation u(x) = v(x) F(d) = F p p ( 2 1 ) a+c: τ=0, θ=0, D u =0.5*10-4, D v =0, D w =10-3, λ=3.0, κ 1 =-0.1, κ 3 =1.0, κ 4 =1.0, Ω=[0,1.2], x=5*10-3, b+d: D u =1.1*10-4, D w =9.64*10-4, λ=1.71, κ 1 =-0.15, x=2.5*10-3, others as in a+c. Liehr 2003 5-4

Determination of the Interaction Function from Experimental Trajectories Using New Stochastic Data Analysis acceleration a / mm s -2 distance d / mm parameters: U 0 =4600 V, ρ SC =3,55 MΩ cm, R 0 =4,4 MΩ, Gas: N 2, T=100 K, p=283 hpa, D=30 mm, d=550 µm, a SC =1 mm, I=233µA, t exp =20 ms, f rep =50 Hz Bödeker et al. 2003 5-5

Examples for the Interaction Behaviour of Dissipative Solitons in R 2 Obtained from the Model Equations reduced equation (a), field equation (b)-(d) interaction diagram trajectories I II ( τ, ξ ) = (3.35,0.02) ( τ, ξ ) = (3.35,0.174) B I white area: scattering : rotating molecules I ξ : parallel offset B II : rotating molecules II τ : relaxation time 6.87 10 = ( ( )) = d 4 15.7d F(d) e cos 43.15 d 0.199, Q 1950 θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[-0.5,0.5]x[-0.5,0.5], x=5*10-3, t=0.1. Liehr et al. 2003 5-6

Phase Separation in the Solutions of the Reduced Equation for many Dissipative Solitons Ν=81 marked solitons in the largest cluster at starting time τ=3.34, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[-2,2]x[-2,2], x=5*10-3, t=0.1; F(d), Q = 1950. Röttger 2003 5-7

Chapter 6 Quantitative Model for Planar Gas-Discharge Systems with High Ohmic Barrier

Experimental Set-Up for Measuring Self-Organized Patterns in Planar DC Gas-Discharge Systems 6-1

tne + div( Γ e) = S e, n + div( Γ ) = S, t p p p 1 ϕ= e(n e n p ), ε0 Γ e = µ enee De n e, Γ =µ n E D n, p p p p p Se = Sp =ν ne β nen p, E = ϕ, µ =µ (E), ν ν=ν (E), e,p e,p De,p = kt e,p µ e,p (E)/e, j = e( Γ Γ ). g p e Model Equation for the DC Gas-Discharge System I: Gas-Discharge Space (-d < z < 0) n e,n p electron/ion density S e, S p electron/ion source term Γ, Γ electron/ion particle current density µ e, µ p electron/ion mobility E electrical field D e, D p ν β ϕ e T e, T p j g p electron/ion diffusion constant ionisation rate recombination rate electrical potential electron/ion temperature global electrical current density Amiranashvili, Purwins 2003 6-2

Model Equation for the DC Gas-Discharge System II: Semiconductor Wafer (0 < z < d SC ) jsc =λ E, E = ϕ, div( ε ϕ ) = 0. j SC λ E ϕ ε global electrical current density specific electrical conductivity electrical field electrical potential dielectric constant of the semiconductor Amiranashvili, Purwins 2003 6-3

Model Equation for the DC Gas-Discharge System III: Boundary Conditions Gas Semiconductor at z=0 1 ε 0 σ t DS σ= e(j z g j SC) z= 0, σ= ( ε ee) (ee), z z =+ 0 z z = 0 1 ( Γ pe) z z = 0 = ( µ pnee p z + np v p ) z = 0, 4 ( Γ ee z) z = 0 = ( µ eneeez 1 + n e v e γγ p e z ) z = 0, 4 v = 8k T / π m. e,p e,p e,p σ surface charge D S e z ε v e,v p γ k diffusion constand of surface charge unity vector in z-direction dielectric constant of the semiconductor thermal electron/ion speed γ -Townsend-coefficient Boltzmann-constant m e,m p electron/ion mass U voltage drop at the component ( ϕ ) ( ϕ ) = U. z = d z = d SC Amiranashvili, Purwins 2003 6-4

Chapter 7 Dissipative Solitons in Various Systems

Experimental Set-Up for Measuring Self-Organized Patterns in Quasi-1-Dimensional DC Gas-Discharge Systems electric circuit Radehaus et al. 1987 7-1

Cascade of Current Filaments in a Quasi-1-Dimensional DC Gas-Discharge System I increasing driving voltage U S Radehaus et al. 1987, Willebrand et al. 1992 7-2

Cascade of Current Filaments in a Quasi-1-Dimensional DC Gas-Discharge System II decreasing driving voltage U S Radehaus et al. 1987, Willebrand et al. 1992 7-3

Travelling and Interacting Current Filaments in a Quasi-1-Dimensional DC Gas-Discharge System increasing driving voltage U S Willebrand et al. 1992 7-4

Brauer, Purwins 2000 7-5 Experimental Set-Up for Measuring Self-Organized Patterns in Planar AC Gas-Discharge Systems

Brauer, Purwins 2000 7-6 Experimentally Observed Hierarchy of Filamentary Patterns in the DBD System I

Brauer, Purwins 2000 7-7 Experimentally Observed Hierarchy of Filamentary Patterns in the DBD System II

Experimentally Observed Hierarchy of Filamentary Patterns in the DBD System III Brauer, Purwins 2000 7-8

Current Filaments in n-gaas Plates Measured at 4.2 K by Electron Microscopy decrease of driving voltage from A to F Mayer et al. 1988 7-9

Clusters of Localized States Measured in a Laser Driven Na Vapour Cell Schäpers et al. 2000 7-10

Keynes 1979 7-11 Nerve Pulse Propagation I: Mechanism for Generation of Electrical Potential Difference

Keynes 1979 7-12 Nerve Pulse Propagation II: Membrane Potential and Ion Conductivity Plotted as a Function of Time