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

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

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

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)

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

[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]

Resonance – Circle of Fixed Points Bh 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]

[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]

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]

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?

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

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

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

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

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]

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

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

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

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

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

Finding Energy Bifurcations Resonance Parabolic GB Parabolic resonance

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

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

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

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.

Regular Energy Levels Solutions ε ~ 10-3

Elliptic Resonance k=2, Ω=1, ε ~ 10-4 i.c. (x, y, I, γ) = (0,0,1,-π)

Hyperbolic Resonance k=1.025, Ω=1, ε ~ 10-4 i.c. (x, y, I, γ) = (1,0,1,-π)

Homoclinic Chaos k=1.025, Ω=1, ε ~ 10-4 i.c. (x, y, I, γ) = (0.895,0,1.4,π)

Parabolic Resonance k =√2, Ω=1, ε ~ 10-4 i.c. (x, y, I, γ) = (0,0,1,-π)

Global Bifurcation k =√3/8 , Ω=1, ε ~ 10-4 i.c. (x, y, I, γ) = (0,0,1,π)

Resonant Global Bifurcation k =1/2 , Ω=1, ε ~ 10-4 i.c. (x, y, I, γ) = (0,0,1,π)

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

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

Thank you!