Mechanisms of chaos in the forced NLS equation

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Mechanisms of chaos in the forced NLS equation

• Eli Shlizerman
• Vered Rom-Kedar
• Weizmann Institute of Science

eli.shlizerman_at_weizmann.ac.il
http//www.wisdom.weizmann.ac.il/elis/
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The autonomous NLS equation

• Boundary
• Periodic B(xL,t) B(x,t)
• Even B(-x,t) B(x,t)
• Parameters
• Wavenumber k 2p / L
• Forcing Frequency O2

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Integrals of motion

• The Particle Number
• The Energy
• The Perturbation

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The problem

• Classify instabilities in the NLS equation

Time evolution near plane wave
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Solitons

• Solitary wave
• Permanent shape B (x , t) g (x)
• Traveling wave solution B (x , t) g (x - vt)
• Localized g (r) 0 r
  ?8
• Particle like
• Preserved under collisions

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Plane wave solution
Bh
Bh
Re(B(0,t))
Re(B(0,t))
?0
?0
Bpw
Bpw
Im(B(0,t))
Im(B(0,t))
Heteroclinic Orbits!
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Modal equations

• Consider two mode Fourier truncation
• B(x , t) c (t) b (t) cos (kx)
• Substitute into the unperturbed eq.

Bishop, McLaughlin, Ercolani, Forest, Overmann
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General Action-Angle Coordinates

• For b?0 , consider the transformation
• Then the systems is transformed to
• We can study the structure of

Kovacic
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The Hierarchy of Bifurcations

• Local Stability for I lt 2k2

Fixed Point Fixed Point Stable Unstable
x0 y0 I gt 0 I gt ½ k2
xx2 y0 I gt ½k2 -
Fixed points in (x,y) are circles in 4
dimensional space
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Perturbed motion classificationnear the plane
wave

• Close to the integrable motion
• Standard dyn. phenomena
• Homoclinic Chaos, Elliptic Circles
• Special dyn. phenomena
• PR, ER, HR

Dashed Unstable Solid Stable
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Analogy between ODE and PDE
ODE
I
H0
BpwPlane wave
PDE
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Analogy between ODE and PDE
ODE
I
H0
PDE
BhHomoclinic Solution
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Analogy between ODE and PDE
ODE
I
H0
PDE
-BhHomoclinic Solution
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Analogy between ODE and PDE
ODE
I
H0
PDE
BsolSoliton (XL/2)
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Analogy between ODE and PDE
ODE
I
H0
BsolSoliton (X0)
PDE
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Numerical simulations - Surface plot
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B plane plot
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EMBD
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I-? plot
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Conclusions

• Three different types of chaotic behavior and
  instabilities in Hamiltonian perturbations of the
  NLS are described.
• The study reveals a new type of behavior near the
  plane wave solution Parabolic Resonance.
• Possible applications to Bose-Einstein
  condensate.

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Characterization Tool

• An input Bin(x,t) can we place this solution
  within our classification?
• Quantitative way for classification
  (tool/measure)
• HC - O(e), HR - O(e1/2), PR - O(e1/3)
• Applying measure to PDE results

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The measure smax
y
x
Measure smax std( B0j max)
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smax PDF for fixed e
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smax dependence on e
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Future Work

• Capturing the system into PR by variation of the
  forcing
• Instabilities in the BEC
• Resonant surface waves

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Thank you!
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Summary

• We analyzed the modal equations with the
  Hierarchy of Bifurcations
• Established the analogy between ODE and PDE
• Numerical simulations of instabilities
• Characterization tool

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Analogy between ODE and PDE
-BsolSoliton (XL/2)
BsolSoliton (X0)
BpwPlane wave
-BhHomoclinic Solution
BhHomoclinic Solution
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The Hierarchy of Bifurcations

• We can construct the EMBD for all fixed points
  in the model

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Previous experiments
D. McLaughlin, K. McLaughlin, Overmann, Cai
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Evenness condition

• Without evenness
• For small L - the solutions are correlated

D. McLaughlin, K. McLaughlin, Overmann, Cai
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Local Stability

• Plane 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 lt k2 lt c2

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Local Stability

• But k is discretized by L so
• kj 2pj/L for j 0,1,2
• (j - number of LUMs)
• Substitute to 0 lt k2 lt c2 and get
• 2pj/L lt c lt 2p(j1)/L
• As we increase the amplitude the number of LUMs
  grows.

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Validity of the model

• For plane wave (b0)
• Substituting the condition for c for 1 LUM
• 2pj/L lt c lt 2p(j1)/L j1
• Then the 2 mode model is plausible for
• I lt 2k2

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Analogy between ODE and PDE

• Constants of motion
• The solution