Mechanisms of chaos in the forced NLS equation

Mechanisms of chaos in the forced NLS equation Eli Shlizerman Vered Rom-Kedar Weizmann Institute of Science eli.shlizerman@weizmann.ac.il http://www.wisdom.weizmann.ac.il/~elis/

The autonomous NLS equation • Boundary • Periodic B(x+L,t) = B(x,t) • Even B(-x,t) = B(x,t) • Parameters • Wavenumberk = 2π / L • Forcing FrequencyΩ2

Integrals of motion • The “Particle Number”: • The “Energy”: • The “Perturbation”:

The problem Classify instabilities in the NLS equation Time evolution near plane wave

Solitons • Solitary wave • Permanent shapeB (x , t) = g (x) • Traveling wave solutionB (x , t) = g (x - vt) • Localized g (r) = 0 r →±∞ • Particle like • Preserved under collisions

Plane wave solution Bh Bh Re(B(0,t)) Re(B(0,t)) θ₀ θ₀ Bpw Bpw Im(B(0,t)) Im(B(0,t)) Heteroclinic Orbits!

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]

General Action-Angle Coordinates • For b≠0 , consider the transformation: • Then the systems is transformed to: • We can study the structure of [Kovacic]

The Hierarchy of Bifurcations Local Stability for I < 2k2 Fixed points in (x,y) are circles in 4 dimensional space

I H0 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

y x Analogy between ODE and PDE ODE I H0 PDE Bpw=Plane wave

y x Analogy between ODE and PDE ODE I H0 PDE +Bh=Homoclinic Solution

y x Analogy between ODE and PDE ODE I H0 PDE -Bh=Homoclinic Solution

y x Analogy between ODE and PDE ODE I H0 PDE +Bsol=Soliton (X=L/2)

y x Analogy between ODE and PDE ODE I H0 PDE +Bsol=Soliton (X=0)

I H0 I H0 I H0 Numerical simulations - Surface plot

B plane plot

EMBD

I-γ plot

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.

Characterization Tool • An input: Bin(x,t) – can we place this solution within our classification? • Quantitative way for classification (tool/measure) HC - O(ε), HR - O(ε1/2), PR - O(ε1/3) • Applying measure to PDE results

The measure: σmax y x Measure:σmax = std( |B0j| max)

σmax PDF for fixed ε

σmax dependence on ε

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

Thank you!

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

I H0 y x Analogy between ODE and PDE -Bsol=Soliton (X=L/2) +Bsol=Soliton (X=0) Bpw=Plane wave -Bh=Homoclinic Solution +Bh=Homoclinic Solution

The Hierarchy of Bifurcations We can construct the EMBD for all fixed points in the model:

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

Evenness condition Without evenness: • For small L - the solutions are correlated D. McLaughlin, K. McLaughlin, Overmann, Cai

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 < k2 <|c|2

Local Stability • But k is discretized by L so kj = 2πj/L for j = 0,1,2… (j - number of LUMs) • Substitute to 0 < k2 < |c|2 and get 2πj/L < |c| < 2π(j+1)/L • As we increase the amplitude the number of LUMs grows.

Validity of the model • For plane wave (b=0): • Substituting the condition for |c| for 1 LUM: 2πj/L < |c| < 2π(j+1)/L j=1 • Then the 2 mode model is plausible for I < 2k2

Analogy between ODE and PDE • Constants of motion • The solution

Mechanisms of chaos in the forced NLS equation