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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

results.jpg

Dynamics of cubic non-linear Schrödinger equation over a 2D square domain.

Animation

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