Instabilities in the Forced Truncated NLS

1. Bh Bh “Special” Dynamical Phenomena Bpw φ₀ Bpw Instabilities in the Forced Truncated NLS E. Shlizerman and V. Rom-Kedar, Weizmann Institute of Science, Israel 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 Plain Wave Solution Two Mode Fourier Truncation • A solution which is independent of X. [8] Substituting in the perturbed (conservative) NLS the approximation [7, 9,10] Leads to a Hamiltonian equation, which is integrable at ε=0. B(x , t) = c (t) + b (x,t) • The 1-D cubic integrable NLS is of the following form: Bpw(0 , t) = |c| e i(ωt+φ₀) (+) focusing Homoclinic Orbits to the Plain Wave Solution Family of homoclinic orbits to the PW exists: (-) de-focusing dispersion Bh(x , t) t±∞Bpw(0 , t) • Small perturbation can be added Forcing Damping + iεαu iεΓ e i(Ω² t+θ) • The forced autonomous equation is obtained by u = B e -i Ω² t[8,11] Generalized Action-Angle Coordinates for c≠0 [6] Resonant Plain Wave Solution c = |c| eiγb = (x + iy)eiγ I = ½(|c|2+x2+y2) Leads to unperturbed Hamiltonian equations with H=H(x,y,I): Whenω=0 circle of fixed points occur [7,11] Bpw(0 , t) = |c| e iφ₀ • Parameters: • Wavenumber k = 2π / L • Forcing Frequency Ω2 Homoclinic Orbits become Heteroclinic orbits! • Conditions: • Periodic u (x , t) = u (x + L , t) • Even Solutions u (x , t) = u (-x , t) Hierarchy of Bifurcations Level 1 - Single energy surface - EMBD, Fomenko Level 3 - Parameter dependence of the energy bifurcation values - k, Ω Fomenko Graphs: (example for line 5) Preliminary Step: Local Stability [6] Example: Parabolic Resonance [1,2,3,5] Parabolic Circle Ip= ½ k2 PR: IR=IPk2=2Ω2 Resonance IR= Ω2 General approach: Fix k and construct H(Ω) diagram Singularity Surfaces Level 2 - Energy bifurcation values - Changes in EMBD EMBD Construction [1,2,4] H4 4 6 5* Changes in the EMBD H1 H3 • Fold - Resonance H2 • Change in stability - Parabolic • Crossing – Possible Global Bifurcation Parameters: k=1.025 , Ω=1 Dashed – Unstable Full – Stable Perturbed Motion Hyperbolic Resonance Close to Integrable and Standard Perturbed Motion Perturbed Motion Classification Close to the integrable motion “Standard” Dynamical Phenomena Homoclinic Chaos, Elliptic Circle k=1.025, Ω=1, ε ~ 10-4 i.c. (x, y, I, γ) = (1,0,1,-π) Parabolic Resonance, Hyperbolic Resonance, etc. Parabolic Resonance 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, 2004. [2] E. Shlizerman and V. Rom-Kedar. Hierarchy of bifurcations in the truncated and forced NLS model. CHAOS,15(1), 2005. [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. k =√2, Ω=1, ε ~ 10-4 i.c. (x, y, I, γ) = (0,0,1,-π)