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
+ (1−iβ)∆Ψ− iδΨ+ (1−iϵ) |Ψ|2Ψ− (1+iμ) |Ψ|4Ψ = 0,
β = 0.05, δ = −0.01, μ = −0.8, ϵ = 0.44.
Rectangle 120×100 (two-dimensional) with its center at the origin.
Mesh grading is 300×300×1 was obtained with use of blockMesh utility.
The function at zero time is radial Gaussian distribution with small mixture of P and D spherical waves,