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. (-) dispersion focusing • A solution which is independent of X. [8] Substituting in the perturbed (conservative) NLS the approximation [7, 9, 10] Bh B(x , t) = c (t) + b (x, t) Leads to a Hamiltonian equation, which is integrable at ε=0. Homoclinic Orbits to the Plain Wave Solution Family of homoclinic orbits to the PW exists: de-focusing Forcing • Small perturbation can be added Two Mode Fourier Truncation Bpw(0 , t) = |c| e i(ωt+φ₀) • The 1 -D cubic integrable NLS is of the following form: (+) The Plain Wave Solution Damping Bh(x , t) t ±∞ Bpw(0 , t) Bpw iεΓ e i(Ω² t+θ) + iεαu • 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| eiγ Bh When ω=0 circle of fixed points occur [7, 11] b = (x + iy)eiγ I = ½(|c|2+x 2+y 2) Leads to unperturbed Hamiltonian equations with H=H(x, y, I): Bpw(0 , t) = |c| e iφ₀ Parameters: Wavenumber Forcing Frequency φ₀ k = 2π / L Ω 2 Homoclinic Orbits become Heteroclinic orbits! Conditions: Periodic u (x , t) = u (x + L , t) Even Solutions u (x , t) = u (-x , t) Bpw Hierarchy of Bifurcations Fomenko Graphs: (example for line 5) Level 1 - Single energy surface - EMBD, Fomenko Level 3 - Parameter dependence of the energy bifurcation values - k, Ω Preliminary Step: Local Stability [6] Fixed Point x=0 Stable y=0 Example: Parabolic Resonance [1, 2, 3, 5] Unstable H(xf , yf , I; k, Ω = const) I>0 I>½ k 2 Parabolic Circle Resonance H 1 x=±x 2 y=0 I > ½k 2 - H 2 x =0 y=±y 3 I > 2 k 2 - y=±y 4 I > 2 k 2 H 4 I R= Ω 2 PR: IR=IP k 2=2Ω 2 H 3 x =±x 4 I p= ½ k 2 - Singularity Surfaces General approach: Fix k and construct H(Ω) diagram Level 2 - Energy bifurcation values - Changes in EMBD Construction [1, 2, 4] H 4 4 5* 6 Changes in the EMBD H 3 H 1 • Fold - Resonance H 2 • Change in stability - Parabolic • Crossing – Possible Global Bifurcation Parameters: k=1. 025 , Ω=1 Dashed – Unstable Full – Stable Perturbed Motion Classification Close to Integrable and Standard Perturbed Motion Hyperbolic Resonance Close to the integrable motion “Standard” Dynamical Phenomena Homoclinic Chaos, Elliptic Circle “Special” Dynamical Phenomena 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. Mc. Laughlin and K. T. R. Mc. Laughlin. The Non. Linear Schrodinger Equation as both a PDE and a Dynamical system. [9] A. R. Bishop, M. G. Forest, D. W. Mc. Laughlin 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. Mc. Laughlin 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)
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