Complex Ginzburg-Landau EquationLecture I

Igor Aranson & Lorenz Kramer, Reviews of Modern Physics,The World of the Complex Ginzburg-Landau Equation,v 74, p 99 (2002)

Tentative plan

• Lecture 1. General Introduction. Real GLE• Lectures 2 & 3. Complex GLE

Complex Ginzburg-Landau Equation

• A(x,y,z,t) –complex amplitude

• Laplace operator

• b – linear dispersion• c – nonlinear dispersion

Preamble

• The complex Ginzburg-Landau equation (CGLE)is one of the most-studied nonlinear equations inthe physics community.

• It describes phenomena from nonlinear waves tosecond-order phase transitions, superconductivity,superfluidity and Bose-Einstein condensation etc.

• Our goal is to give an overview of phenomena in1D, 2D and 3D.

Some Historic Comments

• Vitalii Ginzburg received Nobel Prize inPhysics 2003 for writing the GL equation

• Alex Abrikosov received Nobel Prize inPhysics 2003 for a particular (and incorrect)stationary solution to the GL equation

• What would Landau have said about CGLE?Pathological Science?

Why do we need to study the CGLE?

Definitions

• CGLE describes isotropic extended systems nearthe threshold of long-wavelength supercriticaloscillatory instability (Hopf Bifurcation).

• Near the threshold the equation assumes universalfrom.

• Equation is written in terms of complex amplitudeof the most unstable oscillatory mode.

Examples

• B-Z type chemical reactions (2D, 3D)• Wide-aperture lasers (2D)• Electro-convection in liquid crystals (1D)• Hydrodynamic flows (1D)• Flames (1D, 2D)• Micro-organisms colonies (2D)• System near the threshold of orientation transition

(2D)

Cardiac activity

Self-Assembling Microtubules andMolecular Motors

T. Surray, F. Nedelec, S. Leibler & E. Karsenti, Physical Properties Determining Self-Organizationof Motors & Microtubules, Science, 292 (2001)

Superconductivity and Superfluidity

NOBEL PRIZE WINNER — Argonne physicist Alexei A. Abrikosovhas won the 2003 Nobel Prize for Physics, along with Anthony Leggettof the University of Illinois, Urbana-Champaign,and Vitaly Ginzburg of the P.N. Lebedev Physical Institute in Moscow.

Observed Patterns in the CGLE

Connections to Condensed Matter

• (real) Ginzburg-Landau Equation (b,c=0)Superconductivity, superfluidity near Tc

• Nonlinear Schrödinger Equations Superconductivity, superfluidity for T=0 , nonlinear optics

(fully integrable system in 1D)

History: Hopf Bifurcation (0D)•Poincare considered the problem too trivial to write down….•Andronov & Leontovich did it on the plane ( ~1938).•Hopf generalized it to many degrees of freedom (1942).

|A|=r/g - radius of the cycle, r ~ marg A - angle, w – frequency

Landau or normal form equation (amplitude equation)

u0,v0-steady state solutionAt m=0 real part of 2 complex roots cross zeroPeriodic motion emerges (limit cycle)

Stability of fixed point

•Linear stability analysis

u

v

u

v

m<0

m>0 limit cycle

u0

v0

Re l

Im lw-w

. .

Example: Limit cycle in the van der Pol Equation

Asymptotic method: perturbative solution

Substituting in the first order we obtain linear equation for the correction term

From the condition that no resonance we obtain the Landau equation

Solving the Landau Equation

Polar coordinates A= R ei j

R=1 – stable point, R=0 unstable point. Stable solution is a circle of radius R=1

General case (the complex Landau equation)

Home work: derive the Landau equation from

Generalization to arbitrary order system of eqns

U – vector of dimension N, L – NxN matrix, F – nonlinearity,w – eigenfrequency, U0 – eigenvector

Expansion (transition into a rotating frame)

Solvability conditions (Fredholm alternative)

Zero eigenvector of conjugated system

From linear algebra

Spatially Extended SystemsPhenomenologically: Newell and Whitehead 1970/1971Derived for destabilization of plane Couette flowStewartson & Stuart, DiPrima, Ekhaus and Siegel, 1971

•Amplitude equation should reproduce linear dispersion relationin the long-wavelength limit•Local evolution near the threshold should reproduce Landauequation or normal form (because u(t),v(t) are solutions of full system)

Linear Stability Analysis

Linearized System

Linear Behavior

• for q=0 (Hopf bifurcation)

• for small q – long-wavelength oscillatory instability max growth-rate for homogeneous oscillations

Re(l)

q

Re(l)

qx

qy

Weakly Non-Linear Analysis

CGLE-result of solvability conditions

• Substitute anzatz into original system• Collect terms at O(e3/2)• Orthogonalization with respect to eigenvector (U1,V1)

• D,g can be scaled out

Home work: derive the Ginzburg-Landau equationfrom

Classification of Bifurcation Scenario

• General form of linear growth-rate in generic anisotropicsystems (expanded near critical wavenumber qc and frequencywc)

• vg-linear group velocity, t and x characteristic time and length

Real Ginzburg-Landau Equation

(i) wc=b=0, qc = 0 – real GLE (e.g. 1D)

Examples: stationary bifurcation in Rayleigh-Benard

convection, superconductivity & superfluidity (but fortotally different reason), system near orientationtransition in 2D

Complex Ginzburg-Landau Equation

(ii) wc≠ 0, qc = 0 – classic CGLE Examples: oscillatory chemical reactions, certain class of

wide-aperture lasers, biological systems

Coupled Complex Ginzburg-LandauEquations

(iii) wc≠ 0, qc ≠ 0 – 2 CGLE for counter-

propagating waves A,B Examples: many 1D hydrodynamic and optical systems

Short Wave Instability: Swift-Hohenberg Equation• Linear growth-rate is max at |q|=qc General form of linear

growth-rate in isotropic systems Complex Swift-Hohenberg Equation

qx

qy

Rel(q)

Wensink et al, Mesoscale turbulence in livingfluids, PNAS, 2012

Generic Properties of CGLE

• Translation Invariance: r→r+const• Isotropic: angle q→q+const• Gauge Invariant: A→Aeij, j=const• Inversion in param space: (b,c,A)→(-b,-c,A*)• Hidden symmetries (inherited from the NSE)

Select Solutions of the CGLE

• Plane Waves Solutions: 1D• Vortices and Spirals: 2D• Vortex Filaments or Lines: 3D

Plane-wave solutions

For b=c=0 (real GLE) Vg=w=0

Plane-wave solutions

Plane-wave solutions

For b=c=0 (real GLE) Vg=w=0

Difference with linear wavesLinear waves Nonlinear Waves in the CGLE

Frequency does not depend onamplitude

Frequency is function of amplitudeW= - b q2 – c A2

Amplitude does not depend onwavenumber q

Amplitude depends on thewavenumber A2=1- q2

Waves decay due to dissipation Waves do not decay, system isactive, consumes energy

Waves do not interact, linearsuperposition

Waves interact, nonlinear collisionsand shocks

No frequency selection Topological defects and boundariesselect unique frequency

Linear wave equation

Spiral Solution

F(r)

y(r)

For real GLE y=w=0

3D vortex filaments

Few facts about real GLE• 1D Stationary GLE is fully integrable. We write it for amplitude-phase

variable A=I eij

• “Integral of Motion” from the second equations

• Home work: find the solution to stationary 1D GLE

Vortices in 2D case

Asymptotic Behaviors

for m=±1

F

r

Slowly Drifting Vortex Solution

Derivation of the Drift Velocity

Zero modes of adjoint operator

Consider Nonlinear PDE L(U)=0. Let U0(r) is the solution, L(U0(r))=0 anddL(U0(r))w=0 is corresponding linearized equation. If L(U) does not depend explicitlyon r, then satisfies the linearized equation

Adjoint Eigenfunctions

Equations of motion

A Big Trouble with the Mobility

•The mobility diverges for Rè∞???• The vortex does not move???

Saving the Theory

• Regularization of the Mobility (by LorenzKramer et al, and later by Len Pismen)

• Main issue: moving defects do not perturbthe phase far away from the cores

• One needs more accurate approximation for r→∞

Simple Way:

• Introduce cut-off radius Rc ~1/V

• From balance of main terms

Equation for the phase of A

Regularization of the Mobility

To read more: Len Pismen, Vortices in Nonlinear Fields, Oxford, 1999

Regularized Equation of Motion

•Mobility is velocity-dependent•The dependence is non-analytic•Equations of motion are highly nonlinear

Application to Vortex Interactiony

x

First vortex: x=0,y=0,m=1Second vortex: x=x0,y=0,m=-1

v v

oppositely charged –attractionlikely charged – repulsion