Instabilities in the Forced Truncated NLS E. Shlizerman and V. Rom-Kedar, Weizmann Institute of Science, Israel The Nonlinear Shrödinger Equation dispersion 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. Small perturbation can be added (-) de-focusing (-) de-focusing (+) focusing (+) focusing iεΓ e i(Ω ² t+θ) iεαu ForcingDamping The forced autonomous equation is obtained by u = B e -i Ω² t [8,11] Parameters: Wavenumber k = 2π / L Forcing Frequency Ω 2 Conditions: Periodic u (x, t) = u (x + L, t) Even Solutions u (x, t) = u (-x, t) The Plain Wave Solution A solution which is independent of X. [8] B(x, t) = c (t) + b (x,t) B pw (0, t) = |c| e i(ωt+φ ₀ ) Homoclinic Orbits to the Plain Wave Solution B h (x, t)  t  ±∞ B pw (0, t) B pw BhBh Resonant Plain Wave Solution B pw φ₀φ₀ BhBh Family of homoclinic orbits to the PW exists: When ω=0 circle of fixed points occur [7,11] B pw (0, t) = |c| e iφ ₀ Homoclinic Orbits become Heteroclinic orbits! Two Mode Fourier Truncation Substituting in the perturbed (conservative) NLS the approximation [7, 9,10] Leads to a Hamiltonian equation, which is integrable at ε=0. c = |c| eiγ b = (x + iy)e iγ I = ½(|c| 2 +x 2 +y 2 ) Generalized Action-Angle Coordinates for c≠0 [6] 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 [6] Fixed PointStableUnstableH(x f, y f, I; k, Ω = const) x=0y=0 I > 0I > ½ k 2 H1H1 x=±x 2 y=0 I > ½k 2 -H2H2 x =0y=±y 3 I > 2k 2 -H3H3 x =±x 4 y=±y 4 -I > 2k 2 H4H4 EMBD Construction [1,2,4] H2H2 H1H1 H3H3 H4H4 Parameters: k=1.025, Ω=1 Dashed – Unstable Full – Stable 46 5* Fold - Resonance Changes in the EMBD Change in stability - Parabolic Crossing – Possible Global Bifurcation Parabolic Circle I p = ½ k 2 Example: Parabolic Resonance [1,2,3,5] Resonance I R = Ω 2 PR: I R =I P k 2 =2Ω 2 General approach: Fix k and construct H(Ω) diagram Close to the integrable motion “Standard” Dynamical Phenomena “Special” Dynamical Phenomena Parabolic Resonance, Hyperbolic Resonance, etc. Homoclinic Chaos, Elliptic Circle Close to Integrable and Standard Perturbed Motion Hyperbolic Resonance k=1.025, Ω=1, ε ~ i.c. (x, y, I, γ) = (1,0,1,-π) k =√2, Ω=1, ε ~ i.c. (x, y, I, γ) = (0,0,1,-π) References: [1] E. Shlizerman and V. Rom-Kedar. Energy surfaces and hierarchies of bifurcations - instabilities in the forced truncated NLS, Chaotic Dynamics and Transport in Classical and Quantum Systems. Kluwer Academic Press in NATO Science Series C, [2] E. Shlizerman and V. Rom-Kedar. Hierarchy of bifurcations in the truncated and forced NLS model. CHAOS,15(1), [3] A. Litvak-Hinenzon and V. Rom-Kedar. Parabolic resonances in 3 degree of freedom near-integrable Hamiltonian systems. [4] A. Litvak-Hinenzon and V. Rom-Kedar. On Energy Surfaces and the Resonance Web. [5] V.Rom-Kedar. Parabolic resonances and instabilities. [6] G.Kovacic and S. Wiggins. Orbits homoclinic to resonances, with application to chaos in a model of the forced and damped sine-Gordon equation. [7] G.Kovacic. Singular Perturbation Theory for Homoclinic Orbits in a Class of Near-Integrable Dissipative Systems. [8] D. Cai, D.W. McLaughlin and K. T.R. McLaughlin. The NonLinear Schrodinger Equation as both a PDE and a Dynamical system. [9] 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. [10] 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. [11] G. Haller. Chaos Near Resonance. Fomenko Graphs: (example for line 5) Perturbed Motion Perturbed Motion Classification The 1-D cubic integrable NLS is of the following form: Hierarchy of Bifurcations Parabolic Resonance Singularity Surfaces Leads to unperturbed Hamiltonian equations with H=H(x,y,I): +