Outline

1. Outline We examine the existence of parabolic resonances and various regimes of instabilities in the perturbed Nonlinear Shrödinger equation (NLS). • Two-mode Fourier truncation of the NLS pde • Rescaled Model • Homoclinic Structures • Hyperbolic Resonances in the NLS Model: • Energy-momentum bifurcation diagram (EMBD) • Fomenko graphs Tools: • Stability analysis • Identifying parabolic resonances • Global bifurcation Summary:

2. The Nonlinear Shrödinger equation • The Nonlinear Shrödinger (NLS) equation is used as a robust model for nonlinear dispersive wave propagation in widely different physical contexts. It plays an important role in nonlinear optics, waves in water, atmosphere and plasma. • The 1-D cubic integrable NLS is of the following form: focusing dispersion A solitary traveling nonlinear wave solution (soliton) arises in nonlinear systems when a balance between dispersion and focusing (the non-linear term) exists. The soliton is an envelope pulse along the fiber and a good candidate for information bit.

3. Understanding NLS dynamics One of the approaches for understanding the dynamics of the NLS pde is to consider the two mode Fourier truncation, as a reduction of the system to a near integrable two degree of freedom (d.o.f) Hamiltonian model. Note: NLS can be regarded as approximation of the Sine-Gordon Equation (SGE) at low amplitudes. The two-mode Fourier truncation To develop such truncation, NLS approximation for the weekly perturbed SGE, is considered. Substituting Fourier solution to the perturbed NLS approximation and choosing ε=0 for the unperturbed model

4. Finally, the perturbed Hamiltonian model is received by adding small conservative perturbation. To apply methods and techniques for two d.o.f systems, transformation to action-angle coordinates - x,y,I,θ and rescaling-β can be found. Received is the truncated unperturbed model in the form of ODEs k – wave number β – time scaling

5. Finally, the perturbed Hamiltonian model is received by adding small conservative perturbation. ! The two mode model supplies qualitative understanding of the full model, yet, rigorous results for the PDE must use different techniques.

6. Hyperbolic Resonances • Hamiltonian system is in resonance when where is a fixed point in the (x,y) plane. • Homoclinic hyperbolic resonances occur when a system possessing a fixed point, corresponding to a normally hyperbolic circle of fixed points with a family of heteroclinic orbits connecting points on the circle, is perturbed. Hyperbolic resonance (I=Ir) fourth dimensional unperturbed phase space is presented as:

7. The existence of homoclinic orbits was shown, when a • method for proving the existence of homoclinic orbits in a class • of perturbed integrable two d.o.f freedom Hamiltonian systems • was applied for the NLS truncated system. • Near resonant homoclinic structures generating a chaotic behavior were also observed in the experiments performed on the NLS equation.

8. Parabolic Resonances Parabolic resonances is an exclusive type of chaotic behavior of a near integrable Hamiltonian system. It occurs when a parabolic invariant circle of fixed points is perturbed. Parabolic invariant circles appear generically in integrable nonseparable two d.o.f Hamiltonian systems. When additional degeneracy – resonance – occurs, the Hamiltonian system exhibits parabolic resonance. IPR = IP = IR I=Ip In a flat parabolic resonance case, when additional degeneracy appears, large scale and fast instabilities are generated. Large instabilities were observed in the near flat parabolic case for the atmospheric model and appear in some common two d.o.f physical models. ! I=IR

9. Methods and Tools The Energy Momentum Bifurcation Diagram (EMBD) Energy bifurcation diagrams are constructed to understand the possible structures of the energy surfaces of a two d.o.f Hamiltonian system. Using the diagram for the unperturbed system, we can: • Identify singular surfaces which divide between different types of motions. • Find the region of allowed motion. • Find resonant singular surfaces. • Observe Global Bifurcations Some possible energy momentum bifurcation diagrams for two d.o.f Hamiltonian system [3] Elliptic 2-tori , Hyperbolic 2-tori, Parabolic Resonances [1] Elliptic 2-tori [2] Elliptic 2-tori , Hyperbolic 2-tori (dashed)

10. Allowed region of motion for the perturbed orbits, is in a band around the unperturbed singular surfaces. Fomenko Graphs • Using Fomenko Graphs we are able to identify topologically equivalent integrable two d.o.f Hamiltonian systems and classify them. • For a two d.o.f Hamiltonian we can construct a molecule consisting of atoms representing the singular iso-energy non resonant surfaces. It was shown that iso-energy surfaces (H=const) are equivalent when their molecules are equivalent. Perturbed motion Hyperbolic resonance Elliptic resonance Fomenko Graphs Hyperbolic Resonance band Hyperbolic Resonance band width

11. Fomenko graphs can be seen in the Figure above, when: Elliptic singular surface: Hyperbolic singular surface:

12. Singular surfaces of the NLS model Local Stability Analysis • We would like to evaluate the Hamiltonian along the singular surfaces (i.e. along the various fixed points in the (x,y) plane). • It will allow us to see the effect of the rescaling on the curvature of the various surfaces. • We begin by calculating fixed points of the rescaled system in the (x,y) plane and their stability.

13. EMBD of the NLS model • We proceed to evaluating the Hamiltonian for each singular surface, construct the EMBD and corresponding Fomenko graphs. 1. EMBD for β=1 (not rescaled) 2. EMBD for Rescaled System, β=√2 3. EMBD for Rescaled System β=2, Near flat parabolic resonances

14. Parabolic Resonances • Parabolic circles of fixed points (Parabolic Resonances) occur when a fixed point is parabolic and resonant, i.e. for values I=IP=IR.

15. From EMBD construction and the analysis, the system posses two possible values for parabolic resonances: • Instabilities • At the parabolic resonant points instabilities occur. A near integrable Hamiltonian system exhibits large instabilities when additional degeneracy occur. The curvature of the various branches at the parabolic resonant points determines instabilities‘ intensity. • Strong instabilities appear in the Flat Parabolic Resonances case, • when the curvature: is small or approaches to 0. • It can be observed, that in the NLS model fixed points curvature depends only on the parameter β- the time rescaling parameter. To identify strong instabilities we fix the value of k and aim to choose the smallest curvature. • Parabolic resonances will occur when IPR = IR. • In the figures above IPR = 1. IPR1 =2k2β4 IPR2 =1/2k2β4 !

16. Preliminary results

17. References: [1] A. Litvak-Hinenzon and V. Rom-Kedar. Parabolic resonances in 3 degree of freedom near-integrable Hamiltonian systems. [2] A. Litvak-Hinenzon and V. Rom-Kedar. On Energy Surfaces and the Resonance Web. [3] V.Rom-Kedar. Parabolic resonances and instabilities. [4] G.Kovacic and S. Wiggins. Orbits homoclinic to resonances, with application to chaos in a model of the forced and damped sine-Gordon equation. [5] G.Kovacic. Singular Perturbation Theory for Homoclinic Orbits in a Class of Near-Integrable Dissipative Systems. [6] D. Cai, D.W. McLaughlin and K. T.R. McLaughlin. The NonLinear Schrodinger Equation as both a PDE and a Dynamical system. [7] A.R. Bishop, M.G. Forest, D.W. McLaughlin and E.A. Overman II. A Modal Representation of Chaotic Attractors For the Driven, Damped Pendulum Chain. [8] A.R. Bishop, M.G. Forest, D.W. McLaughlin and E.A. Overman II. A quasi-periodic route to chaos in a near-integrable pde. [9] G. Haller. Chaos Near Resonance.