1. 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
3. ES & RK, Energy surfaces and Hierarchies of bifurcations - Instabilities in the
forced truncated NLS, Cargese-03
SNOWBIRD, 2005

2. Near-integrable NLS Conditions Parameters
(+) focusing
dispersion
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
Should I write? ux (0 , t) = 0

3. [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]

4. 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]

5. [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]

6. 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]

7. Then the 2 mode model is plausible for I < 2k2
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
Then the 2 mode model is plausible for I < 2k2

8. 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

9. 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]

10. 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
[Litvak-Hinenzon & RK]

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

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

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

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

15. Finding Energy Bifurcations
Resonance
Parabolic GB
Parabolic resonance

16. 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

17. 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

18. 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

19. Homoclinic Chaos Model PDE
k=1.025, Ω=1, ε ~ i.c. (x, y, I, γ) = (0,0,1.5,π/2)

20. Hyperbolic Resonance Model PDE
k=1.025, Ω2=1, ε ~ i.c. (x, y, I, γ) = (0,0,1,π/2)

21. Parabolic Resonance Model PDE
k=1.025, Ω2=k2/2, ε ~ i.c. (x, y, I, γ) = (0,0,k2/2,π/2)

22. Measure: σmax = std( |B0j| max)
Classification y x
Measure: σmax = std( |B0j| max)

23. Measure Dependence on ε
p is the power of the order: O(εp)

24. Discussion Solutions close to HR Stability of solutions
Applying measure to PDE results
Change red points to some other color