1. Instabilities in the Forced Truncated NLS
Eli Shlizerman
Under the supervision of Prof. Vered Rom-Kedar
Dept. of Comp. Science and Applied Math.
Weizmann Institute of Science

2. Outline Introduction Our work Conclusions + Future work
A framework of hierarchy of bifurcations
Classification of temporal chaos
Numerical simulations
Conclusions + Future work

3. General NLS
The Nonlinear Shrödinger (NLS) equation is used as a robust model for nonlinear dispersive wave propagation in widely different physical contexts.
(+) focusing
dispersion
(-) de-focusing

4. Near-integrable NLS Near integrable Conditions iεΓ e i(Ω² t+θ) iεαu
Forcing
Damping
Near integrable
Should I write? ux (0 , t) = 0
Conditions
Periodic Boundary u (x , t) = u (x + L , t)
Even Solutions u (x , t) = u (-x , t)

5. The Autonomous Equation
Choose u = B * e -i Ω² t
Parameters
Wavenumber k = 2π / L
Forcing Frequency Ω2

6. [McLaughlin, Cai, Shatah]
Homoclinic Orbits Bh
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)
Experiments, Homoclinic orbits to any solution, multi-pulse homoclinic orbits
FIX: the red line inside the blue circle -OK
Add: resonant circle
Bpw
[McLaughlin, Cai, Shatah]

7. Resonance – Circle of Fixed PointsBh
When ω=0 – circle of fixed points occur
Bpw(0 , t) = |c| e i(φ₀)
Heteroclinic Orbits!
Bpw φ₀
Experiments, Homoclinic orbits to any solution, multi-pulse homoclinic orbits
FIX: the red line inside the blue circle -OK
Add: resonant circle
[Haller, Kovacic]

8. [Bishop, McLaughlin, Forest, Overman]
Two Mode Model
Consider two mode Fourier truncation
B(x , t) = c (t) + b (t) cos (kx)
Substitute into the unperturbed eq.:
Add perturbation – OK?
Two mode model approximates the pw solution and cos is supposed to model the hom. structure
Plain wave stability
[Bishop, McLaughlin, Forest, Overman]

9. General Action-Angle Coordinates for c≠0
Consider the transformation:
c = |c| eiγ b = (x + iy) eiγ I = ½(|c|2+x2+y2)
Plain wave stability, perturbation?
[Kovacic]

10. General Action-Angle Coordinates for b≠0
Consider the transformation:
c = (u + iv) eiθ b = |b| eiθ I = ½(|b|2+u2+v2)
Plain wave stability, perturbation?

11. Plain Wave Stability 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

12. Plain Wave Stability – cont.
But k is discretized by L so
kj = 2πj/L for j = 0,1,2… (number of LUMs)
Substitute to 0 < k2 < |c|2 and get
2πj/L < |c| < 2π(j+1)/L
Use for PW (b=0):
I = ½ ( |c|2 + |b|2 )
Validity: say that when I < 2k^2 the phenomena probably occurs in pde and otherwise it is doubtful.
So the 2 mode model (LUM=1) valid for I < 2k2

13. Why do we need Hierarchy of Bifurcations?
Global overview of the integrable structure
Classification of the perturbed motion
Parameters regime
Maybe it should be the summary

14. Hierarchy 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

15. Preliminary step - Local Stability
B(x , t) = [|c| + (x+iy) coskx ] eiγ
Fixed Point
Stable
Unstable
x=0
y=0
I > 0
I > ½ k2
x=±x2
I > ½k2 -
x =0
y=±y3
I > 2k2
x =±x4
y=±y4
Remember stability analysis
[Kovacic & Wiggins]

16. Level 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 H2
x =0
y=±y3 H3
x =±x4
y=±y4 H4

17. EMBD Parameters: k=1.025 , Ω=1 Dashed – Unstable Full – Stable H4 H1

18. Fomenko Graphs and Energy Surfaces
Example: H=const (line 5)

19. Level 2: Energy Bifurcation Values* 4 5* 6
Yellow square around 5 - OK

20. Possible Energy Bifurcations
Folds - Resonances
Unbounded surfaces
Crossings – Global Bifurcation
Branching surfaces – Parabolic Circles H I

21. Finding Energy Bifurcations
Resonance
Parabolic GB
Parabolic resonance

22. What happens when energy 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)
Change red points to some other color
Parabolic Resonance: IR=IP k2=2Ω2

23. Level 3: Bifurcation Parameters
Example of a diagram:
Fix k
Find Hrpw(Ω)
Find Hppw(Ω)
Find Hrpwm(Ω)
Plot H(Ω) diagram
Add slide on PR – when bifurcations coincide

24. Perturbed motion classification
Close to the integrable motion
“Standard” dyn. phenomena
Homoclinic Chaos, Elliptic Circles
Special dyn. phenomena
PR, ER, HR, GB-R
Change red points to some other color

25. Results Presentation Comparison to the integrable motion
Perturbed Energy Surfaces - (I,x,y) or (I,u,v)
Phase diagrams - (I,γ) or (I,θ)
Perturbed motion on the EMBD - (H0,I)
Reconstructed solution
Amplitude plots - (|B(X,t)|, X, t)
B-plane plots - (Re{B(0, t)}, Im{B(0, t)})
First let us consider the representations which will allow an effective comparison between the
perturbed and the underlying unperturbed structures.

26. Regular Energy Levels Solutions
ε ~ 10-3

27. Elliptic Resonance
k=2, Ω=1, ε ~ i.c. (x, y, I, γ) = (0,0,1,-π)

28. Hyperbolic Resonance
k=1.025, Ω=1, ε ~ i.c. (x, y, I, γ) = (1,0,1,-π)

29. Homoclinic Chaos
k=1.025, Ω=1, ε ~ i.c. (x, y, I, γ) = (0.895,0,1.4,π)

30. Parabolic Resonance
k =√2, Ω=1, ε ~ i.c. (x, y, I, γ) = (0,0,1,-π)

31. Global Bifurcation
k =√3/8 , Ω=1, ε ~ i.c. (x, y, I, γ) = (0,0,1,π)

32. Resonant Global Bifurcation
k =1/2 , Ω=1, ε ~ i.c. (x, y, I, γ) = (0,0,1,π)

33. Conclusions Two main themes Hierarchy of bifurcations
Global analysis of a truncated NLS model
General framework for analysis
Hierarchy of bifurcations
First and second levels
Third level of Hierarchy – reveals parameters
We showed new types of perturbed orbits

34. Future Work Comparison with PDE solutions
Tools for a precise distinction between various types of trajectories
Dissipative perturbation
Spatio-temporal chaos
BEC

35. Thank you!