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Gross-Pitaevsky equation, Schrödinger version, small frequency
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
e
^{ia0 x2}
at
t
=0.
Equation to solve
i
∂Ψ(
x
,
y
)
∂
t
+ β
⎛
⎝
1
2
∆Ψ(
x
,
y
) −
C
_{4}
p
^{2}
|Ψ(
x
,
y
)|
^{2}
Ψ(
x
,
y
)
⎞
⎠
= 0
where
p
= [1+2
a
_{0}
(β
_{0}
t
+ β
_{1}
sin(Ω
t
)/Ω)]
^{−1}
, β = β
_{0}
+ β
_{1}
cos(Ω
t
).
Parameters were chosen as
β
_{0}
=0, β
_{1}
=1,
C
_{4}
=−1,
a
_{0}
=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
cosh
x
.
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
Evolution and modulation instability of extended soliton for Gross-Pitaevsky/Schrödinger equation.
Animation
Links related to this case
descript.pdf
: PDF version of this web page
descript.tex
: LaTeX description
beta2m-3a.avi
(54.8 MB): Animation file
results.flv
(4.8 MB): Flash animation file
blockMeshDict
: Mesh generating file
initCosh1.C
: Initial condition: C program
transportProperties
: Initial condition: Input parameters
beta2m.C
: C program
transportProperties
: Input parameters
controlDict
: Time and data input/output control
beta2m.out
: C program: Output
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:
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