Stability in presence of cubic non-linearity in two dimensions
Introduction
Initial conditions are chosen as one-dimensional (no x-dependence. Since a one-dimensional soliton does not collapse, the pattern stays stable in y-direction, until instabilities in x-direction break translational symmetry. Since in two dimensions solitons exhibit self-focusing and collapse, the pattern finally diverges.
Equation to solve
i
∂Ψ(x,y)
∂t
+ ∆Ψ(x,y) + (|Ψ(x,y)|^{2})Ψ(x,y) = 0.
Domain
Square 24×24 (a two-dimensional area with coordinates (x,y)).
Mesh
Mesh grading is 512×512×1 was obtained with use of blockMesh utility.
Initial condition
The function at zero time is Gaussian distribution along y direction, with small modulations along x direction,
Ψ(x,y,z) = e^{−[(y2)/16]} [1+ 0.001sin(πx/3)].
Boundary conditions
Neumann (zero gradient normal to the boundary).
Calculations
The time step is ∆t=2·10^{−4}. Calculation time is around 15 hours.
Results
Dynamics of cubic non-linear Schrödinger equation over a 2D square domain.