Complex Behavior in Coupled Nonlinear Waveguides. Roy Goodman, New Jersey Institute of Technology
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1 Complex Behavior in Coupled Nonlinear Waveguides Roy Goodman, New Jersey Institute of Technology
2 Nonlinear Schrödinger/Gross-Pitaevskii Equation i t = r + V (r) ± Two contexts for today: Propagation of light in a nonlinear waveguide (x, z) gives the electric field envelope Evolution occurs along axis of waveguide ( transverse spatial dimension Potential represents waveguide geometry t! z ) plus one Evolution of a Bose-Einstein condensate (BEC) Everyone s favorite nonlinear playground. A new state of matter achieved experimentally in the 99 s. One, two, or three space dimensions Potential represents magnetic or optical trap
3 Periodic and chaotic tunneling in a 3-well waveguide Why three wells? x z (aka t) Other work on two-waveguide arrays shows symmetry-breaking bifurcations and an associated wobbling dynamics. Three waveguides provide the simplest system in which Hamiltonian Hopf bifurcations, which lead to complex dynamics, are possible. Significant interest in many-waveguide arrays. Useful to proceed: Simple Geometry Complex Geometry, Simple Dynamics Complex Dynamics
4 What got me thinking: Double well Stationary (a).4. Time-dependent dynamics.8.6 (b) V (x) =V (x + L)+V (x L) Experiment in Bose-Einstein condensate L norm N.5 Power Diagram (c) (d) Ψ Ω Ω Frequency (a) Mode shape 5 5 x Spontaneous symmetry (.3) breaking above critical intensity that is found analytically. Kirr, Kevrekidis, Shlizerman, Weinstein 8 see also Fukuizumi & Sacchetti β (b) d....3 c (a) a. (d) b α Figure 4. Anumericalplotofanoscillatorysolutionto(.)attimesa, b, c, and d respectively from the corresponding phase plane diagram of the finite dimensional Hamiltonian truncation. Rigorous F j = πresult: j F, long-time for j =, where (.4) (c) Time dependent dynamics in a single or double well shadowing of ODE solutions by F = [ c ψ + c ψ +(c ] c + c c )ψ ψ R + [ c ψ + c ψ +c ] PDE c ψ ψ R + [ c ψ solutions + c ψ ] R +[c ψ +c ψ ] R + R R, 7 Marzuola & Weinstein Pelinovsky & Phan Goodman, Marzuola, Weinstein 5 Albiez et al. 5
5 What got me thinking: Triple well 3-well potential & eigenfunctions 5 5 V(x) (x), = (x), =. 3 (x), 3 = 9 d dt Periodic Schrödinger Trimer (Johansson J. Phys. A 4) n + C( n n + n+ )+ n n = Real part Imaginary part subject to n+3 = n Hamiltonian Hopf Bifurcations M Johansson Bifurcations of standing waves L -norm (Kapitula/Kevrekidis/Chen SIADS 6) N Odd solutions Even Solutions Asymmetric Solutions Ω Mode unstable for range of N ψ n 3 ψ n ω l /C (a) n= n= n=3-5 - ψ n Λ/C Numerically-generated chaos time time (c) n= n= n= Figure. time 8 6 (b) n= n= n=3 Hamiltonian Hopf bifurcations in the discrete nonlinear Schrödinger trimer 9 Typical instability-generated dynamics for randomly perturbed unstable stationary
6 Two goals Understand what takes place at HH bifurcation as paradigm for nonlinear wave oscillatory instability. (c) Im(λ) (d ) () () (3) (3) () () Im(λ) Re(λ Re(λ) Flesh out the dynamics of relative periodic orbits in the system. Eventual Goal: Which of these dynamics can we prove exist?
7 Finite dimensional reduction Decompose the solution as Z = c (t) (t)+c (t) (t)+c 3 (t) 3(t)+ (x, t) projection onto eigenmodes (x, t)? j (x) Ignoring contribution of (x, t) gives finite-dimensional Hamiltonian system with (approximate) Hamiltonian h H = c + c + 3 c 3 3 A c + c 3 + c 4 +4 c c 3 c + c + c 3 (c c 3 + c c 3 )+ 3 c c 3 + c c 3 + (c 3 c ) c i +( c 3 c ) c For well-separated potential wells, the spectrum has the form with
8 Symmetry reduction System conserves squared L norm N Reduces # of degrees of freedom from 3 to Removes fastest timescale H R =( + ) z +( + ) z 3 AN z + z + z3 + z 3 (z z 3 + z z 3 ) 4(z z 3 + z z 3 ) h A z 4 + z z 3 z (z z 3 + z z 3)+ z + z 3 i 5(z z 3 + z z 3 ) + (z z 3 + z z 3 ) z z z3 z 3. Relative fixed points in full system fixed points in reduction Relative periodic orbits periodic orbits At, semisimple double frequency i = ±i. When >, non-simple double eigenvalues at N HH A and N HH, with instability in between. A
9 Menagerie of standing waves Three branches continue from linear system Six branches arise in saddlenode bifurcations Four stabilizations/ z /N destabilizations in HH bifurcations (a) N (+++) (++) (++) (+) (+) (-++) (-+) (-+-) (-+)
10 More about this picture z /N (a) (+++) (++) (++) (+).3 (+) N (-++) (-+) (-+-) (-+) Lyapunov Center Theorem: (Roughly) For each pair of imaginary eigenvalues of a fixed point, excepting resonance, there exists a oneparameter family of periodic orbits that limits to that fixed point.
11 Bifurcations in Hamiltonian systems change the topology of Lyapunov branches of periodic orbits Standard Example: Hamiltonian Pitchfork ẍ = x + x 3 (a) (b) β β α α > <
12 ODE & PDE simulations Real(λ) Trivial solution stable N x 3 (a), N=.35 5 x 4 Re(σ ) Y 75 5 t X x 4 Real(z ) Poincaré Section (t)
13 ODE & PDE simulations Real(λ) Chaotic heteroclinic bursting N (c), N=... Re(σ ) Y. 4 t...4 X Real(z ) Poincaré Section (t)
14 Reduced Hamiltonian has 4 daunting terms! H R =( + ) z +( + ) z 3 AN z + z + z3 + z 3 (z z 3 + z z 3 ) 4(z z 3 + z z 3 ) h A z 4 + z z 3 z (z z 3 + z z 3)+ z + z 3 i 5(z z 3 + z z 3 ) + (z z 3 + z z 3 ) z z z3 z 3. Goal: understand periodic orbits of H R using Hamiltonian Normal Forms Given a system with Hamiltonian H = H (z)+ H(z, ) find a near-identity canonical transformation z = F(y, ) such that the transformed Hamiltonian K(y, ) =H (F(y, ), )=H (y)+ K(y, ) is simpler than H(z, ).
15 What does simpler mean? Try to remove terms from H to construct K Eliminating terms at a given order in terms of higher order introduces new A term can be removed if it lies in the range of the adjoint ad H = {,H } operator of. Invoke Fredholm alternative. Resonant terms in adjoint null space. Project Hamiltonian onto this subspace. For { example } in our problem i / 3 z z 3 z 3 z z z 3 (a) Degree Two / 3 z z 3 z z z 3 z 3 z z 3 z z 3 z 3 4 z 3 z z 3 z 4 z z z 3 z z 3 (b) Degree Four
16 Three normal form calculations Real(λ) HH HH Semisimple -: resonance for Gives HH at N crit = A + O( ) Nonsemisimple -: resonance at N crit using a further simplification of above normal form Nonsemisimple -: resonance computed numerically at numerical location of HH N, N= O( )
17 Normal form near semisimple double eigenvalue (Chow/Kim 988) Normal Form H = z + z 3 Real(λ) N H norm = z + z 3 + z + z 3 +AN(z z 3 + z z 3 ) apple + A z 4 z z 3 + z 3 4 z + z 3 (z z 3 + z z 3 ) In Canonical Polar Coordinates H = ( J + J 3 )+ (J + J 3 )+4AN p J J 3 cos ( + 3 ) + A J J J 3 + J 3 4 p J J 3 (J + J 3 ) cos ( + 3 ) Independent of ( 3 ) implying the existence of a conserved quantity and the integrability of the Normal Form. Advantage: Easier to find solution structure in Normal Form.
18 The system can be further reduced. Periodic orbits J e i t solve: p J J 3 ( A (J + J 3 )) + A N (J + J 3 ) J 6J J 3 J 3 cos = p J J 3 (N J J 3 )sin = J 3 With =( + 3 ) 5 J and J3 act as barycentric coordinates on the 5 J + J 3 = N triangle of admissible J 3 solutions showing relative strength of the three modes. J
19 Sequence of bifurcations in Normal Form J3 J 3 J 3 J 3 J J J J Lyapunov families of fixed points + unphysical branch Unphysical branches cross into physical region Lyapunov branches pinch off Question: At second bifurcation point HH, must have Lyapunov families of fixed point. Where do they come from?
20 Normal form for non-semisimple -: resonances at HH and HH (Meyer-Schmidt 974) In symplectic polar coordinates (r,,p r,p ), this is: H =H (r, p r,p ) +µ H (r, p ) +H 4 (r, p ) = p + p r + p r +µ ap + b r + c p + d p r + e 8 r4 = ±, µ Poincaré-Lindstedt argument: periodic orbits with amplitude µr and frequency + µ! when there is a solution to! er = Hyperbolic (a) b< Elliptic e > e < δ< δ= δ> Two cases: b> (b) b> ω ω
21 The bifurcation at HH Real(λ) Frequency Period Computations using previous normal form ω J +J 3 Amplitude Period ω J +J 3 ω J +J Increasing N Numerically Computed Periodic orbits (not normal form) (a).5. (x () + x 3 () )/ Period (b) (x () + x 3 () )/ Period (c) (x () + x 3 () )/ N
22 Some computed PDE solutions on this branch 75 (b) B 7 Period 65 C 6 A (x () + x 3 () )/
23 r The bifurcation at HH Period Period 9 8 Numerically Computed Periodic orbits y + y 3 (a) Increasing N Amplitude ODE Computation N=.43 N=.45 N= y (b) PDE Computation -.. y Real(λ) N New family of periodic orbits arises in elliptic HH bifurcation.3.. (b) (b) Period (a) y N=.45 N= N= y + y y ω
24 Increasing N Solutions must satisfy z + z 3 <N. PDE.8.6 y 3 ().4. (a) y 3 () (b) ODE N =.8 N = y () y ()
25 What s going on? Getting close to other fixed points z /N (a) (+++) (++) (++) (+) N = N (+) (-++) (-+) (-+-) (-+) What about the other Lyapunov branches of periodic orbits?
26 I thought saddle-node bifurcations were boring.9.8 (a) (+++) (++) Saddle-node z /N (++) (+).3 (+) Saddle-node N (-++) (-+) (-+-) (-+)
27 Normal form for bifurcation i! Small beyond all orders remainder H = q + p Fast & Oscillatory + p + q q 3 3 Saddle node bifurcation at = + Iq +H higher (q,i)+r (q,q,p,p ) Coupling where I = q + p Three families of periodic orbits: Fast Slow Mixed
28 Perturbation expansion shows two regimes N<N crit N>N crit Saddle-node.4 (c). Saddle-node q Mixed periodic orbits bifurcate when = 4 n q
29 Saddle-node x 3 () (a) N=.496 x 3 () (b) N=.496 x 3 () (c) N= x () x () x () (d) N=.99 (e) N=.34. (f) N= x 3 () -. x 3 () -. x 3 () x () x () x () Gelfreich-Lerman 3
30 Saddle-node (a) N= (b) N= x 3 ().74.7 x 3 () x () x ().8 (c) N= x 3 () x ()
31 A menagerie of periodic orbits computed by continuation
32 Parting Words This problem has an ODE part and a PDE part Increasing from two wells to three makes the ODE part of the problem hard In addition to standing waves, there is a whole lot of additional structure in solutions that oscillate among the three waveguides Normal forms give us a partial picture of the reduced dynamics Even saddle-node bifurcations are interesting. Big question: What can be proven about shadowing these orbits in NLS/GP? For re/preprints
33 Thanks!
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