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Gross-Pitaevsky equation, Schrödinger version, small frequency

Introduction

Coefficients of the equation depend on time. It has reportedly an analytic solution that is equal to 1/cosh x eia0 x2 at t=0.

Equation to solve


i ∂Ψ(x,y)

t
+ β
1

2
∆Ψ(x,y) − C4 p2 |Ψ(x,y)|2Ψ(x,y)
= 0
where
p = [1+2 a00 t+ β1 sin(Ωt)/Ω)]−1,    β = β0+ β1 cos(Ωt).
Parameters were chosen as
β0=0,    β1=1,    C4=−1,    a0=0,    Ω = 0.1

Domain

Cube 10.2×10.2×10.2 centered at the origin.

Mesh

Mesh grading 100×100×100 was obtained with use of blockMesh utility.

Initial condition

The function at zero time is one-dimensional (planar) soliton extended in y and z directions, with small modulations along y and z directions,
Ψ(x,y) = (1+ 0.001 sin z) 1

coshx
.

Boundary conditions

Neumann (zero gradient normal to the boundary) in y direction, and periodic (cyclic) boundary condition in x direction.

Calculations

The time step is ∆t=0.001. Calculation time is around 18 hours, up to t=32.

Results

results.jpg

Evolution and modulation instability of extended soliton for Gross-Pitaevsky/Schrödinger equation.

Animation

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