numerical studies of collapses in ... -...
TRANSCRIPT
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
ExtraNumerical Studies of
Collapses in Turbulencein Nonlinear Schrodinger Equation
N. Vladimirova and P. Lushnikov
University of New Mexico
May 20, 2012
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
Nonlinear Schrodinger equation (NLS)
iψt +∇2ψ + |ψ|2ψ = 0
iψt + (1− iεa)∇2ψ + (1 + iεc)|ψ|2ψ = iεbψ
application: propagation of light through nonlinear media;ψ(x , y , t) — envelope of electric field
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
pdf of |ψ|... side note
pdf of collapses
collapse similarity
F (hmax) → pdf (ψ)
Collapses
Extra
Distribution of |ψ| in the field
Notations: h ≡ |ψ|, havg = 〈|ψ|〉 ∝√
N, N =∫|ψ|2d2r .
0
0.2
0.4
0.6
0.8
1
0 1 2 3
P(h
)
h / havg
b = 20, ε=0.01
c=0, a= 1 c=1, a=1/4c=1, a= 1 c=1, a= 2 c=2, a= 1 2x exp(-x2)
1e-12
1e-08
1e-04
1e+00
1 10 100
P(h
)
h / havg
h-5
h-6
c=0, a= 1 c=1, a=1/4c=1, a= 1 c=1, a= 2 c=2, a= 1
Turbulent background:
I pdf scales with havg
I universal shape for h ∼ havg
Tails:
I power-law (?) for h�havg
I depend on a, c , not on b
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
pdf of |ψ|... side note
pdf of collapses
collapse similarity
F (hmax) → pdf (ψ)
Collapses
Extra
Other field quantities scale with havg
Spatial correlation length (∼ havg ) ←Correlation time (∼ h2
avg ) ← depend on a, c , not on b
Spectra of |ψ|2k (∼ havg ) ←
When rescaled to havg ,
The shape of spatial correlation function is universal.The shapes of temporal correlation function and spectra
(slightly) depend on a, c .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
pdf of |ψ|... side note
pdf of collapses
collapse similarity
F (hmax) → pdf (ψ)
Collapses
Extra
Frequency of collapses with h > hmax.
1e-04
1e-03
1e-02
1e-01
0 100 200 300 400 500
F(h
max
)
hmax
c = 1, b = 20a= 1/4a= 1/2a= 1 a= 2
1e-04
1e-03
1e-02
1e-01
0 100 200 300 400 500F
(hm
ax)
hmax
a = 1, b = 20c= 0 c= 1/4c= 1/2c= 1 c= 2
I Frequency of collapses with h > hmax per unit area: F (hmax).
I The dependence is exponential.
I The exponent depends on a, c , and b.
I No obvious scaling with havg .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
pdf of |ψ|... side note
pdf of collapses
collapse similarity
F (hmax) → pdf (ψ)
Collapses
Extra
Universality of collapses in turbulence: a=1, c=0
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
pdf of |ψ|... side note
pdf of collapses
collapse similarity
F (hmax) → pdf (ψ)
Collapses
Extra
Universality of collapses in turbulence: a=1, c=0
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
pdf of |ψ|... side note
pdf of collapses
collapse similarity
F (hmax) → pdf (ψ)
Collapses
Extra
Universality of collapses in turbulence: a=1, c=0
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
pdf of |ψ|... side note
pdf of collapses
collapse similarity
F (hmax) → pdf (ψ)
Collapses
Extra
Universality of collapses in turbulence: a=1, c=0
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
pdf of |ψ|... side note
pdf of collapses
collapse similarity
F (hmax) → pdf (ψ)
Collapses
Extra
Connecting PDF of |ψ| to frequency of collapses
1e-03
1e-02
1e-01
0 50 100 150
F(h
max
)
hmax
c = 1, b = 20 a = 1/4a = 1/2a = 1 a = 2
1e-03
1e-02
1e-01
0 50 100 150F
(hm
ax)
hmax
a = 1, b = 20 c = 1/4c = 1/2c = 1 c = 2
I assume similarity among collapses, h ∼ (tc − t)−12
I assume known frequency of collapses above hmax: Fmax(hmax)
I conclude that PDF in the field P(h) ∼ h−5Fmax(h)
I Solid lines: F (hmax). Points: 0.012 h5P(h).
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
Diagnostics of collapses
1.8
2
2.2
2.4
2.6
-1.5 -1 -0.5 0 0.5 1
V0
β
I In slow variables ρ = rL and τ =
∫L−2(t ′)dt ′
ψ(r , t) = 1LV (ρ, τ)e iτ+iγ(τ)ρ2/4 (1)
critical NLS becomes
∇2V + V 3 − V + 14βρ
2V = 0. (2)
I Solve (2) for each β, tabulate V0(β) at the center.
I Taylor expansion of (2) at the center and (1) give
L = L ( |ψ|(0)|, |ψ|′′rr (0) ) ⇒ V0 ⇒ β; γ = 2L2φ′′rr (0)
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
Diagnostics of collapses
1.8
2
2.2
2.4
2.6
-1.5 -1 -0.5 0 0.5 1
V0
β
I In slow variables ρ = rL and τ =
∫L−2(t ′)dt ′
ψ(r , t) = 1LV (ρ, τ)e iτ+iγ(τ)ρ2/4 (1)
critical NLS becomes
∇2V + V 3 − V + 14βρ
2V = 0. (2)
I Solve (2) for each β, tabulate V0(β) at the center.
I Taylor expansion of (2) at the center and (1) give
L = L ( |ψ|(0)|, |ψ|′′rr (0) ) ⇒ V0 ⇒ β; γ = 2L2φ′′rr (0)
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
Collapse growth and decay
10
100
1000
-0.02 -0.01 0 0.01
h
t - tc
εa = 1e-4εa = 8e-5εa = 6e-5εa = 0
10
100
1000
1e-05 1e-04 1e-03 1e-02 1e-01h
|t - tc|
|t-tc|-1/2
|t-tc|-1
Evolution of collapses with h0 = 2.8, r0 = 1 with c = 0, b = 0.Growth rate depends on a and c , decay rate does not.Notation: h ≡ |ψ|r=0.
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
Rescaled evolution: (a + 2c) similarity
0
50
100
150
200
250
-0.004 -0.002 0 0.002
h
t - tc
a = 2, c = 0c = 1, a = 0
0
0.2
0.4
0.6
0.8
1
-200 -100 0 100h
/ hm
ax
(t - tc) h2max
a = 2, c = 0c = 1, a = 0a + 2c = 0.2
Left:NLS coordinatescollapses with ε = 0.01, a + 2c = 2, and different h0.
Right:coordinated rescaled with hmax;same curve for collapses with a + 2c = 2,different for a + 2c = 0.2
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
Maximum height of collapses: (a + 2c) similarity
ε = 0.01
0
50
100
150
3 3.5 4 4.5
h max
/ h 0
h0
a+2c
=0.
3
a+2c
=1
a+2c
=1.5
a+2c=2
a+2c=3
a = 0c = 0other
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
Growth rate of collapses: (a + 2c) similarity
1
2
3
3.63 4
1 102 104 106
h (t
c -
t)1/
2
(tc - t) h2max
ε(a+2c) = 1e-4ε(a+2c) = 2e-4ε(a+2c) = 5e-4ε(a+2c) = 1e-3ε(a+2c) = 2e-3
0.04
0.06
0.08
0.10
0.12
0.002 0.005 0.01 0.02 0.04
α
ε(a + 2c)
slope = 0.5
h ∝ (tc − t)−( 12 +α), α = α(a + 2c)
Left: h(t), compensated by (tc − t)−12 .
Right: coefficient α determined from dotted lines on the left panel.
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
ODE model for collapse growth(Fibich and Levy, 1998)
L2βt = −k1 − k2β + k3(LLt)2 − ν(β)
L3Ltt = −β
ν(β) = 45.1e−π/√
|β|
k1 = 2ε(a + 2c)N
M
k2 = εaP − N
M
k3 = εaP − N − 2M
M
N = 2π
ZρR2dρ ≈ 11.69
P = 2π
Zρ3R2
ρdρ ≈ 18.65
M =π
2
Zρ3R2dρ ≈ 3.46
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
Test of ODE model: term-by-term comparison
-0.008
-0.006
-0.004
-0.002
0
0.002
0.01 1 100 10000
L2 βt +
k1
(tc - t) h2max
solid: k1 = ε(a+2c) 2N/ Mdashed: k1 = 0
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015
-0.06 -0.04 -0.02 0 0.02 0.04 0.06L2 β
t + k
1
β
k2 = -37ε(a+2c)
εa=1e-4εa=8e-5εa=6e-5εa=4e-5εa=2e-5ν(β)
-k2β
L2βt = −k1 − k2β + k3(LLt)2 − ν(β)
L3Ltt = −β
Data suggest: k1 term describes first order effects well;k2 ≈ −37ε(a + 2c) instead of original k2 ≈ 2εa.
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
Side note on ODE model
0
0.2
0.4
0.6
0.8
1
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
h / h
max
β
εa=1e-4εa=8e-5εa=6e-5εa=4e-5εa=2e-5
0
0.2
0.4
0.6
0.8
1
-2 -1 0 1 2
h / h
max
β k1-2/3
h/hmax = 1/cosh(x+1)
In slow variables keep first two terms of ODE model:
βτ = −k1 − k2β
Lττ − 2L2τ/L = −βL
Results in hhmax
= LmaxL
≈ cosh−1h−
“β − βmax
”|βmax|
12
i,
where β = βk− 2
31 and βmax = −0.92
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
diagnostics
growth and decay
rescaled h(t)
maximum height
growth rate
ODE model
ODE terms
...side note
ODE test
Extra
Test of ODE model: collapse evolution
0
100
200
300
400
-0.002 -0.001 0 0.001
h
t - tc
ε(a+2c) = 2e-4
NLS ODE, hic = 30FL, hic = 30FL, hic = 100
-0.04
-0.02
0
0.02
-0.002 -0.001 0 0.001β
t - tc
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Parameter space: a, c , and b (ε = 0.01).
0
0.5
1
1.5
2
0 0.5 1 1.5 2h a
vg( N / area )1/2
0.85 xc=0, a=1 c=1, a=1 c=1, b=20a=1, b=20c=0, b=30
0
1000
2000
3000
0 10 20 30 40
N
b
N/area = 0.09 b
I Notations: h ≡ |ψ|, havg = 〈|ψ|〉, N =∫|ψ|2d2r .
I No obvious scalings with a, c , or b......except N ∝ b and havg ∝
√N.
I We will show that havg well describes background turbulence.
I What determines havg?
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Distribution of |ψ| in the field scales with havg
0
0.2
0.4
0.6
0.8
1
0 1 2 3
P(h
)
h / havg
b = 20, ε=0.01
c=0, a= 1 c=1, a=1/4c=1, a= 1 c=1, a= 2 c=2, a= 1 2x exp(-x2)
1e-12
1e-08
1e-04
1e+00
1 10 100P
(h)
h / havg
h-5
h-6
c=0, a= 1 c=1, a=1/4c=1, a= 1 c=1, a= 2 c=2, a= 1
I PDF of |ψ| scales with havg .
I Universal shape for h ∼ havg , power-law (?) tails for h�havg .
I The tails do not depend on b.
I The tails do depend on a and c .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Distribution of |ψ| in the field scales with havg
1e-12
1e-10
1e-08
1e-06
1e-04
1e-02
1e+00
1 10 100
P(h
)
h / havg
c = 0, a = 1
b = 30b = 20b = 10
1e-12
1e-10
1e-08
1e-06
1e-04
1e-02
1e+00
1 10 100P
(h)
h / havg
c = 1, a = 1
b = 30b = 20b = 10
I PDF of |ψ| scales with havg .
I Universal shape for h ∼ havg , power-law (?) tails for h�havg .
I The tails do not depend on b.
I The tails do depend on a and c .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Distribution of |ψ| in the field scales with havg
1e-12
1e-10
1e-08
1e-06
1e-04
1e-02
1e+00
1 10 100
P(h
)
h / havg
c = 1, b = 20
a = 1/4a = 1/2a = 1 a = 2
1e-12
1e-10
1e-08
1e-06
1e-04
1e-02
1e+00
1 10 100P
(h)
h / havg
a = 1, b = 20
c = 0 c = 1/4c = 1/2c = 1 c = 2
I PDF of |ψ| scales with havg .
I Universal shape for h ∼ havg , power-law (?) tails for h�havg .
I The tails do not depend on b.
I The tails do depend on a and c .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Spatial correlation function scales with havg
0
0.2
0.4
0.6
0.8
1
0 1 2
S2
r havg
c = 0, a = 1
b = 30b = 20b = 10Gauss
0
0.2
0.4
0.6
0.8
1
0 1 2S
2
r havg
c = 1, a = 1
b = 30b = 20b = 10Gauss
I Spatial correlation function scales with havg .
I Correlation length does not depend on b.
I Correlation length depend on a and c .
I The shape, when rescaled to correlation length, is universal.
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Spatial correlation function scales with havg
0
0.2
0.4
0.6
0.8
1
0 1 2
S2
r havg
c = 1, b = 20
a = 1/4a = 1/2a = 1 a = 2
0
0.2
0.4
0.6
0.8
1
0 1 2S
2
r havg
a = 1, b = 20
c = 0 c = 1/4c = 1/2c = 1 c = 2
I Spatial correlation function scales with havg .
I Correlation length does not depend on b.
I Correlation length depend on a and c .
I The shape, when rescaled to correlation length, is universal.
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Temporal correlation function scales with h2avg
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1t h2
avg
c = 0, a = 1
b = 30b = 20b = 10
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1t h2
avg
c = 1, a = 1
b = 30b = 20b = 10
I Temporal correlation function scales with h2avg .
I Correlation time does not depend on b.
I Correlation time depend on a and c .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Temporal correlation function scales with h2avg
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1t h2
avg
c = 1, b = 20
a = 1/4a = 1/2a = 1 a = 2
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1t h2
avg
a = 1, b = 20
c = 1/4c = 1/2c = 1 c = 2
I Temporal correlation function scales with h2avg .
I Correlation time does not depend on b.
I Correlation time depend on a and c .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Energy spectra scale with havg
1e-16
1e-14
1e-12
1e-10
1e-08
1e-06
1e-04
1e-02
0.1 1 10 100
| ψk|
2
k / (2π havg)
c = 0, a = 1
-5b = 30b = 20b = 10
1e-16
1e-14
1e-12
1e-10
1e-08
1e-06
1e-04
1e-02
0.1 1 10 100| ψ
k|2
k / (2π havg)
c = 1, a = 1
-6b = 30b = 20b = 10
I Spectra of |ψ|2k scale with havg .
I The tails do not depend on b.
I The tails do depend on a and c .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Energy spectra scale with havg
0
0.001
0.002
0.003
0.004
0.005
0 0.2 0.4 0.6 0.8 1
| ψk|
2
k / (2π havg)
c = 0, a = 1b = 30b = 20b = 10
0
0.001
0.002
0.003
0.004
0.005
0 0.2 0.4 0.6 0.8 1| ψ
k|2
k / (2π havg)
c = 1, a = 1b = 30b = 20b = 10
I Spectra of |ψ|2k scale with havg .
I The tails do not depend on b.
I The tails do depend on a and c .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Energy spectra scale with havg
1e-16
1e-14
1e-12
1e-10
1e-08
1e-06
1e-04
1e-02
0.1 1 10 100
| ψk|
2
k / (2π havg)
c = 1, b = 20
-6a = 1/4a = 1/2a = 1 a = 2
1e-16
1e-14
1e-12
1e-10
1e-08
1e-06
1e-04
1e-02
0.1 1 10 100| ψ
k|2
k / (2π havg)
a = 1, b = 20
-5c = 1/4c = 1/2c = 1 c = 2
I Spectra of |ψ|2k scale with havg .
I The tails do not depend on b.
I The tails do depend on a and c .
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Test: critcal collapses
1
10
100
1000
1e-05 1e-04 1e-03 1e-02 1e-01
|ψm
ax|
tc - t
h0 = 2.76h0 = 2.80h0 = 2.85
0.001
0.01
0.1
1
1e-05 1e-04 1e-03 1e-02 1e-01L
tc - t
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Test: critcal collapses
0
0.05
0.1
0.15
1e-04 1e-03 1e-02 1e-01
β
tc - t
-L3 Ltt
(tc - t)2 L4/4
-0.3
-0.2
-0.1
0
1e-04 1e-03 1e-02 1e-01
γ
tc - t
LLt h0 = 2.76h0 = 2.80h0 = 2.85
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Test: critcal collapses
-0.0003
-0.0002
-0.0001
0
0.07 0.071 0.072 0.073
L2 βt
β
h0 = 2.80-0.65 ν(β)
-0.003
-0.002
-0.001
0
0.09 0.1 0.11 0.12L2 β
t
β
h0 = 2.85-0.50 ν(β)
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Beta and gamma with stabilization
10
100
1000
-0.02 -0.01 0 0.01
h
t - tc
εa = 1e-4εa = 8e-5εa = 6e-5εa = 0
10
100
1000
1e-05 1e-04 1e-03 1e-02 1e-01h
|t - tc|
|t-tc|-1/2
|t-tc|-1
Evolution of collapses with h0 = 2.8, r0 = 1 with c = 0, b = 0.
Here, ε(a + 2c) = 0.01. Notice that (a+2c) similarity is moreprononeced in L and its derivatives rather than in β and γ.
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Beta and gamma with stabilization
-0.05
0
0.05
0.1
1e-05 1e-04 1e-03 1e-02 1e-01
β
tc - t
- L3 Ltt
-0.3
-0.2
-0.1
0
1e-05 1e-04 1e-03 1e-02 1e-01
γ
tc - t
εa = 1e-4εa = 8e-5εa = 6e-5εa = 0
LLt
Evolution of collapses with h0 = 2.8, r0 = 1 with c = 0, b = 0.
Here, ε(a + 2c) = 0.01. Notice that (a+2c) similarity is moreprononeced in L and its derivatives rather than in β and γ.
Collapses inNLS Turbulence
N. Vladimirova andP. Lushnikov
NLS
Turbulence
Collapses
Extra
parameter space
pdf of psi
spatial corr fun
time corr fun
spectra
critial collapse
beta and gamma
Beta and gamma with stabilization
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.1 1 10 100 1000
β
(tc - t) h2max
β, a = 0-L3Ltt, a = 0β, c = 0
-L3Ltt, c = 0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.1 1 10 100 1000
γ
(tc - t) h2max
γ, a = 0LLt, a = 0γ, c = 0LLt, c = 0
Evolution of collapses with h0 = 2.8, r0 = 1 with c = 0, b = 0.
Here, ε(a + 2c) = 0.01. Notice that (a+2c) similarity is moreprononeced in L and its derivatives rather than in β and γ.