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

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

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

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

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

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

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

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

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

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]

EMBDEMBD

Parameters: k=1.025 , Ω=1

H2

H1

H3

H4

Dashed – Unstable

Full – Stable

Fomenko Graphs and Energy Fomenko Graphs and Energy SurfacesSurfaces

Example: H=const (line 5)

Level 2: Energy Bifurcation ValuesLevel 2: Energy Bifurcation Values

4 65**

Possible Energy BifurcationsPossible Energy Bifurcations Branching surfaces – Parabolic Circles Crossings – Global Bifurcation Folds - Resonances

H

I

0I

H0θ

pI

31 HH

Finding Energy BifurcationsFinding Energy Bifurcations

Resonance Parabolic GB

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

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

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

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

Hyperbolic ResonanceHyperbolic Resonance

k=1.025, Ω2=1, ε ~ 10-4 i.c. (x, y, I, γ) = (0,0,1,π/2)

Model

PDE

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

ClassificationClassification

x

y

Measure: σmax = std( |B0j| max)

Measure Dependence on Measure Dependence on εε

p is the power of the order: O(εp)

DiscussionDiscussion

Solutions close to HR

Stability of solutions

Applying measure to PDE results