Damped Free Propagation

In this example, we will simulate a damped free propagation. We will define a simple 2D system with a Gaussian initial condition and a damping term in the dispersion relation to simulate the damping effect.

We first load the necessary packages.

using GeneralizedGrossPitaevskii, CairoMakie, StructuredLight

Then, we define the parameters of the grid for our simulation and the initial condition.

L = 5
lengths = (L, L)
tspan = (0, 1)
N = 128
rs = StepRangeLen(0, L / N, N)
u0 = (ComplexF64[exp(-(x - L / 2)^2 - (y - L / 2)^2) for x in rs, y in rs],);

Then, we define the dispersion relation with a damping term.

dispersion(ks, param) = sum(abs2, ks) / 2 - im;

This corresponds to an equation of the form i ∂u(r, t)/∂t = -∇²u / 2 - i u, which includes a damping term due to the imaginary part.

Now we can create the problem instance.

prob = GrossPitaevskiiProblem(u0, lengths; dispersion);

Now, we define the solver parameters, get the solution and visualize the results.

dt = 0.02
nsaves = 64
alg = StrangSplitting()

ts, sol = solve(prob, alg, tspan; dt, nsaves, show_progress=false);

save_animation(abs2.(sol[1]), "free_prop_example_damp.mp4", share_colorrange=true);

We can see that the wavefunction is damped over time, which is expected due to the damping term in the dispersion relation.

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