instabilities in the forced truncated nls
DESCRIPTION
Instabilities in the Forced Truncated NLS. Eli Shlizerman and Vered Rom-Kedar Weizmann Institute of Science. 1. ES & RK, Characterization of Orbits in the Truncated NLS Model, ENOC-05. 2. ES & RK, Hierarchy of bifurcations in the truncated and forced NLS model, CHAOS-05. - PowerPoint PPT PresentationTRANSCRIPT
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Instabilities in the Forced Instabilities in the Forced Truncated NLSTruncated NLS
Eli Shlizerman and Vered Rom-KedarWeizmann Institute of Science
SNOWBIRD, 2005
1. ES & RK, Characterization of Orbits in the Truncated NLS Model, ENOC-05 2. ES & RK, Hierarchy of bifurcations in the truncated and forced NLS model, CHAOS-05
3. ES & RK, Energy surfaces and Hierarchies of bifurcations - Instabilities in the forced truncated NLS, Cargese-03
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Near-integrable NLSNear-integrable NLS
Conditions Periodic Boundary u (x , t) = u (x + L , t) Even Solutions u (x , t) = u (-x , t)
Parameters Forcing Frequency Ω2
Wavenumber k = 2π / L
ixxt eiBBBiB )( 22
(+) focusing
dispersion
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Homoclinic OrbitsHomoclinic Orbits
For the unperturbed eq. B(x , t) = c (t) + b (x,t)
Plain Wave Solution Bpw(0 , t) = |c| e i(ωt+φ₀)
Homoclinic Orbit to a PW Bh(x , t) t±∞ Bpw(0 , t)
Bpw
[McLaughlin, Cai, Shatah]
Bh
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Resonance – Circle of Fixed PointsResonance – Circle of Fixed Points
When ω=0 – circle of fixed points occur Bpw(0 , t) = |c| e i(φ₀)
Heteroclinic Orbits!
[Haller, Kovacic]
Bpw
φ₀
Bh
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Two Mode ModelTwo Mode Model Consider two mode Fourier truncation
B(x , t) = c (t) + b (t) cos (kx)
Substitute into the unperturbed eq.:
2222222224224 cb+c b8
1 |c|
2
1-|b|k+Ω
2
1-|b|
16
3|c||b|
2
1|c|
8
1
=H
[Bishop, McLaughlin, Forest, Overman]
=I
)|b||c(|2
1 22 )cΓ(c2
εiH *
p
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General Action-Angle Coordinates General Action-Angle Coordinates for cfor c≠≠00
Consider the transformation: c = |c| eiγ b = (x + iy) eiγ I = ½(|c|2+x2+y2)
[Kovacic]
3y41y2x
43y2kx
2xy433x
47x2I2ky
2xI2Ωγ
0Iγ
H
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Plain Wave StabilityPlain Wave Stability
Then the 2 mode model is plausible for I < 2k2
Plain wave: B(0,t)= c(t)
Introduce x-dependence of small magnitude B (x , t) = c(t) + b(x,t)
Plug into the integrable equation and solve the linearized equation. From dispersion relation get instability for:
0 < k2 < |c|2
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Hierarchy of BifurcationsHierarchy of Bifurcations
Level 1 Single energy surface - EMBD, Fomenko
Level 2 Energy bifurcation values - Changes in EMBD
Level 3 Parameter dependence of the energy
bifurcation values - k, Ω
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Preliminary step - Local StabilityPreliminary step - Local Stability
Fixed Point Stable Unstable
x=0 y=0 I > 0 I > ½ k2
x=±x2 y=0 I > ½k2 -
x =0 y=±y3 I > 2k2 -
x =±x4 y=±y4 - I > 2k2
[Kovacic & Wiggins]
B(x , t) = [|c| + (x+iy) coskx ] eiγ
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Level 1: Singularity SurfacesLevel 1: Singularity Surfaces
Construction of the EMBD -(Energy Momentum Bifurcation Diagram)
Fixed Point H(xf , yf , I; k=const, Ω=const)
x=0 y=0 H1
x=±x2 y=0 H2
x =0 y=±y3 H3
x =±x4 y=±y4 H4
[Litvak-Hinenzon & RK]
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EMBDEMBD
Parameters: k=1.025 , Ω=1
H2
H1
H3
H4
Dashed – Unstable
Full – Stable
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Fomenko Graphs and Energy Fomenko Graphs and Energy SurfacesSurfaces
Example: H=const (line 5)
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Level 2: Energy Bifurcation ValuesLevel 2: Energy Bifurcation Values
4 65**
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Possible Energy BifurcationsPossible Energy Bifurcations Branching surfaces – Parabolic Circles Crossings – Global Bifurcation Folds - Resonances
H
I
0I
H0θ
pI
31 HH
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Finding Energy BifurcationsFinding Energy Bifurcations
Resonance Parabolic GB
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What happens when energy What happens when energy bifurcation values coincide?bifurcation values coincide?
Example: Parabolic Resonance for (x=0,y=0)
Resonance IR= Ω2
hrpw = -½ Ω4
Parabolic Circle Ip= ½ k2
hppw = ½ k2(¼ k2 - Ω2)
Parabolic Resonance: IR=IP k2=2Ω2
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Level 3: Bifurcation ParametersLevel 3: Bifurcation Parameters
Example of a diagram:
Fix k
Find Hrpw(Ω)
Find Hppw(Ω)
Find Hrpwm(Ω)
Plot H(Ω) diagram
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Perturbed motion classification Perturbed motion classification
Close to the integrable motion
“Standard” dyn. phenomena Homoclinic Chaos, Elliptic Circles
Special dyn. phenomena PR, ER, HR, GB-R
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Homoclinic ChaosHomoclinic Chaos
k=1.025, Ω=1, ε ~ 10-4 i.c. (x, y, I, γ) = (0,0,1.5,π/2)
Model
PDE
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Hyperbolic ResonanceHyperbolic Resonance
k=1.025, Ω2=1, ε ~ 10-4 i.c. (x, y, I, γ) = (0,0,1,π/2)
Model
PDE
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k=1.025, Ω2=k2/2, ε ~ 10-4 i.c. (x, y, I, γ) = (0,0,k2/2,π/2)
Model
PDE
Parabolic ResonanceParabolic Resonance
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ClassificationClassification
x
y
Measure: σmax = std( |B0j| max)
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Measure Dependence on Measure Dependence on εε
p is the power of the order: O(εp)
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DiscussionDiscussion
Solutions close to HR
Stability of solutions
Applying measure to PDE results