Solutions of the ComplexGinzburg Landau Equation

1
Solutions of the Complex-Ginzburg Landau Equation
John Pate University of Arizona
Two Dimensional t2
t60 y x t170
t300 Note the amplitude colors are not
the same scale for all plots.

• Project Description
• Look at solutions, particularly traveling wave
  and traveling hole solutions, to the complex
  Ginzburg-Landau equation (CGL) and try to
  understand the behavior of those solutions.
• Applications
• The CGL is an envelope equation which has
  applications to 3
• Chemical reactions
• Liquid crystals
• Hydrodynamic convection
• Nonlinear optics
• Pattern formation

• Traveling hole solutions
• An analytic expression for traveling hole
  solutions was found by Nozaki and Bekki 4.
  Objects reminiscent of traveling holes are seen
  in simulations of CGL in the Benjamin-Feir
  unstable regime.

• Numerical Algorithm
• Start with initial conditions A(0,x)
• Take the Fourier transform in space (this is
  natural because we have found periodic solutions)
• After taking the Fourier transform,
  differentiation with respect to space is the same
  as multiplication in Fourier space. This reduces
  our PDE into an infinite number of ODEs (one for
  each wave number).
• Advance in time while in Fourier space using ODE
  techniques
• This is where an approximation is going to have
  to be made. For the solutions below, the
  nonlinear terms were assumed constant at the
  scale of each time step. Using Taylor series, it
  is not difficult to derive higher order methods
  of integration (Trapezoidal rule, Simpsons rule,
  etc).
• Take the inverse Fourier transform to recover the
  solution in Euclidean space
• Repeat
• Exact formulas for the integration can be found
  in 1.

Figure 2. Computed on 0,50x0,50 with 27 grid
points (same density in both cases). Step size
of 0.05. Solutions were computed using a similar
numerical scheme but with a 2-D FFT and IFFT. In
2-D, what appear to be traveling holes at the
level of the amplitude are actually spiral waves.
Note the colors represent amplitude, and the
scale of the colors is not uniform between
graphs.
Discussion The 1-D and 2-D simulations shown
here illustrate the same phenomenon. Perturbation
of a plane wave solution in the Benjamin-Feir
unstable regime leads to traveling hole solutions
in 1D and spiral waves (or traveling hole at the
level of the amplitude) in 2D. The number of
holes or defects first increases, then saturates.
Numerical Solutions Both the 1-D and 2-D
simulations were ran with 1ab-0.1 One
Dimensional

• References
• Cox, S. and Matthews, P. Exponential Time
  Differencing for Stiff Systems. Journal of
  Computational Physics. 176, 430-455 (2002).
• Lega, J. and Fauve S. Traveling hole solutions to
  the complex Ginzburg-Landau equation as
  perturbations of nonlinear Schrodinger dark
  solutions. Physica D. 102, 234-252 (1997).
• Lega, J. Traveling hole solutions of the complex
  Ginzburg-Landau equations a review. Physica D.
  152, 269-287 (2001).
• Nozaki, S. and Bekki, K. Exact Solutions of the
  Generalized Ginzburg Landau Equation. Journal of
  the Physical Society of Japan. 53, 1581-1582
  (1984).
• http//math.arizona.edu/jpate

Particular Analytic Solutions
Figure 1. Computed on 0,300 with 29 grid
points. Step size of 0.05. Solutions were
computed using the first order convergent Fourier
scheme discussed above.
2. Traveling wave solutions We look for
solutions of the form
by substituting, and we
see this is a solution if and
.
Acknowledgments This project was mentored by
Joceline Lega, whose help is acknowledged with
great appreciation. This material is based upon
work supported by the National Science Foundation
under Grant No. DMS-0405551
Benjamin-Feir Instability By adding a complex
perturbation to the traveling wave solution and
analyzing the equation in Fourier space, we see
the traveling wave solution is unstable if 1ab
2k²(1b²)/(m-k²)lt0. The term involving k is
always positive, so the condition of instability
reduces to 1ablt0.