Complex Ginzburg-Landau Equation Lecture 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 in the physics community. It describes phenomena from nonlinear waves to second-order phase transitions, superconductivity, superfluidity and Bose-Einstein condensation etc. Our goal is to give an overview of phenomena in 1D, 2D and 3D.
Some Historic Comments Vitalii Ginzburg received Nobel Prize in Physics 2003 for writing the GL equation Alex Abrikosov received Nobel Prize in Physics 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 near the threshold of long-wavelength supercritical oscillatory instability (Hopf Bifurcation). Near the threshold the equation assumes universal from. Equation is written in terms of complex amplitude of 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 and Molecular Motors T. Surray, F. Nedelec, S. Leibler & E. Karsenti, Physical Properties Determining Self-Organization of Motors & Microtubules, Science, 292 (2001)
Superconductivity and Superfluidity NOBEL PRIZE WINNER Argonne physicist Alexei A. Abrikosov has won the 2003 Nobel Prize for Physics, along with Anthony Leggett of 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). u0,v0-steady state solution At m=0 real part of 2 complex roots cross zero Periodic motion emerges (limit cycle) Landau or normal form equation (amplitude equation) A =r/g - radius of the cycle, r ~ m arg A - angle, w frequency
Linear stability analysis Stability of fixed point Re l - w.. w Im l u m< 0 u m> 0 limit cycle u 0 v 0 v v
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 Systems Phenomenologically: Newell and Whitehead 1970/1971 Derived for destabilization of plane Couette flow Stewartson & Stuart, DiPrima, Ekhaus and Siegel, 1971 Amplitude equation should reproduce linear dispersion relation in the long-wavelength limit Local evolution near the threshold should reproduce Landau equation or normal form (because u(t),v(t) are solutions of full system)
Linear Stability Analysis
Linearized System
Linear Behavior Re(l) for q=0 (Hopf bifurcation) q for small q long-wavelength oscillatory instability Re(l) max growth-rate for homogeneous oscillations q y q x
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 equation from
Classification of Bifurcation Scenario General form of linear growth-rate in generic anisotropic systems (expanded near critical wavenumber qc and frequency wc) 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 for totally different reason), system near orientation transition 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-Landau Equations (iii) wc 0, qc 0 2 CGLE for counterpropagating 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 Rel(q) Wensink et al, Mesoscale turbulence in living fluids, PNAS, 2012 qx qy
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
For b=c=0 (real GLE) Vg=w=0 Plane-wave solutions
Plane-wave solutions
Plane-wave solutions For b=c=0 (real GLE) Vg=w=0
Linear wave equation Difference with linear waves Linear waves Frequency does not depend on amplitude Amplitude does not depend on wavenumber q Nonlinear Waves in the CGLE Frequency is function of amplitude W= - b q2 c A2 Amplitude depends on the wavenumber A2=1- q2 Waves decay due to dissipation Waves do not interact, linear superposition No frequency selection Waves do not decay, system is active, consumes energy Waves interact, nonlinear collisions and shocks Topological defects and boundaries select unique frequency
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 and dl(u0(r))w=0 is corresponding linearized equation. If L(U) does not depend explicitly on 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 Lorenz Kramer et al, and later by Len Pismen) Main issue: moving defects do not perturb the 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 Interaction y First vortex: x=0,y=0,m=1 Second vortex: x=x0,y=0,m=-1 v v x oppositely charged attraction likely charged repulsion