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Ginzburg-Landau equation (2D case) for ϵ = 0.55

Introduction

The equation has both gain and loss terms, and a small diffusion term. As a result, integrals of density of probability, momentum, and angular momentum are not conserved. The equation has stable (steady state or rotating) solutions when 0.23 < ϵ < 0.53 (for values of other parameters specified below). For smaller ϵ, the solution disperses and tends to zero, for larger ϵ it expands infinitely.

Equation to solve


i ∂Ψ

t
+ (1−iβ)∆Ψ− iδΨ+ (1−iϵ) |Ψ|2Ψ− (1+iμ) |Ψ|4Ψ = 0,
with
β = 0.05,    δ = −0.01,    μ = −0.8,    ϵ = 0.55.

Domain

Rectangle 120×100 (two-dimensional) with its center at the origin.

Mesh

Mesh grading is 300×300×1 was obtained with use of blockMesh utility.

Initial condition

The function at zero time is radial Gaussian distribution with small mixture of P and D spherical waves,
Ψ(x,y) = C0 e−[(r2)/(2R2)] [1+ 0.04 r eiϕ+ 0.02 r2 e2iϕ],
with
C0=0.87,    R=4.03.

Boundary conditions

Dirichlet (zero at the boundary).

Calculations

The time step is ∆t=0.01. Calculation time up to t=1500 (1.5·105 time steps) is around 7 hours.

Results

results.jpg

Dynamics over a 2D rectangular domain.

Animation

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