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Gross-Pitaevsku equation, large non-linear term, small perturbation

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 =   ⎛


α0

α0−2 a02
 
  1

cosh(τ)
,    β = cos(Ωt)
and
τ = arctanh

a0   ⎛


2

α0
 


+

 

2 α0
 
  sinΩt


.
Parameters were chosen as
C4=−1.5,    a0=0,    Ω = 0.1,    α0=0.1.

Domain

Cube 10.2×10.2×10.2 centered at the origin.

Mesh

Mesh grading 32×32×32 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,z) = (1+ 0.01 sin(πy/5.1))(1+ 0.01 cos(πy/5.1)) 1

coshx
.

Boundary conditions

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

Calculations

The time step is ∆t=0.01. Calculation time is around 2 minutes.

Results

results.jpg

Evolution of planar soliton for Gross-Pitaevsky equation.

Animation

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