Linear stability of small-amplitude torus knot solutions of the Vortex Filament Equation

Transcription

1 Linear stability of small-amplitude torus knot solutions of the Vortex Filament Equation A. Calini 1 T. Ivey 1 S. Keith 2 S. Lafortune 1 1 College of Charleston 2 University of North Carolina, Chapel Hill 2010 AMS Spring Central Section Meeting St. Paul, April 10-11, 2010 Partially supported by NSF grant DMS

2 Outline The Vortex Filament Equation The Model Connection with the Nonlinear Schrödinger Equation Stability of solutions Squared Eigenfunctions Small amplitude torus knots Floquet Spectrum Constructing finite-gap vortex filaments Isoperiodic deformations The cabling theorem Stability of small amplitude torus knots Complex double points and linear instabilities Visualizing linear instabilities: Bäcklund transformations

3 The Vortex Filament Equation (VFE) The VFE (Da Rios 1904) gives an idealized description of the motion of a vortex filament in a perfect fluid due to the local effects of its own vorticity. As the core radius becomes zero, with circulation held constant, the filament is described by a curve of position γ(x, t) in R 3, arclength parameter x, and velocity field: γ t (VFE) γ t = γ x 2 γ x 2 = κb, γ xx γ x directed like the binormal vector B of γ and proportional to its curvature κ.

4 Connection with NLS: the Hasimoto map From vortex filaments to NLS solutions. If a curve γ(x, t) of curvature κ and torsion τ satisfies the VFE, γ t = γ x γ xx, then the complex potential (Hasimoto, 1972) q(x, t) = H(γ) = 1 R 2 κ(x, t) x ei τ ds satisfies the nonlinear Schrödinger equation iq t + q xx + 2 q 2 q = 0. (NLS) For example, a family of translating circles of varying curvatures a is mapped to the family of plane wave potentials q a (t) = a 2 ei(a2 /2)t.

5 Connection with NLS: Inverting the Hasimoto map From NLS potentials to VFE solutions. Given a NLS potential q, compute a fundamental solution matrix Φ of the AKNS system. (Its solvability condition is the NLS equation.) ( ) iλ iq φ x = φ, φ i q iλ t = ( ) i( q 2 2λ 2 ) 2iλq q x 2iλ q q x i(2λ 2 q 2 φ. ) Then for Λ 0 R, the skew-hermitian matrix ( ) 1 dφ Φ iγ1 γ dλ = 2 + iγ 3 λ=λ0 γ 2 + iγ 3 iγ 1 gives a solution γ = H 1 (q) of the VFE (modified by a tangential drift) with curvature κ = q and torsion τ = d dx arg(q) + Λ 0. (Sym, Pohlmeyer.)

6 Stability of solutions A solution γ 0 of the VFE is stable when any solution γ initially close to γ 0 (in an appropriate norm) remains close for all times. (Nonlinear stability.) An easier problem to address is that of linear stability: whether all solutions of the linearized equation remain bounded for all times. For a nonlinear evolution equation u t = K[u], its linearization about a solution u 0 (t) is given by v t = δk[u 0 ]v, where δk[u 0 ] is the linear operator obtained by linearizing K[u] about u 0.

7 Linearizations The linearized VFE about a solution γ 0 is γ 1t = γ 1x γ 0xx + γ 0x γ 1xx, (LVFE) where γ 1 is an arclength-preserving perturbation, that is γ 1x T = 0, with T the unit tangent vector of γ 0. The linearized NLS about a potential q 0 is iq 1t = q 1xx + 4 q 0 2 q 1 + 2q 2 0 q 1. (LNLS) Given the natural frame of γ 0 : U 1 = cos φ N sin φ B, U 2 = T U 1, with φ x = τ, and the natural curvatures: κ 1 = κ cos φ, κ 2 = κ sin φ, then for f x = κ 1 g + κ 2 h (arclength preservation): γ 1 = ft + gu 1 + hu 2 solves the LVFE iff q 1 = g + ih solves the LNLS.

8 Squared Eigenfunctions If φ and φ are solutions of the AKNS system at the same value of λ, then the triplet 1 φ φ 2 (φ 1 φ 2 + φ 1 φ 2 ) f = φ 1 φ1 = g, φ 2 φ2 h satisfies the pair of linear systems f 0 i q iq f g h = 2iq 2i q 2iλ 0 0 2iλ g, h f g h t = It follows that: x 0 (2iλ q + q x ) (2iλq q x ) 2(2iλq q x ) 2(2iλ 2 i q 2 ) 0 2(2iλ q + q x ) 0 2(2iλ 2 i q 2 ) The functions g + h and i(g h) are solutions of the LNLS. f g. h

9 Periodic solutions: the Floquet Spectrum For NLS potentials satisfying q(x + L, t) = q(x, t), their discrete and continuous Floquet spectra are defined as follows: σ d (q) = {λ C φ 0 with φ(x + L, t) = ±φ(x, t)}, σ c (q) = {λ C φ 0 bounded for all x R}, where φ(x, t) solves the AKNS system at (q, λ). A typical Floquet spectrum. simple periodic points multiple periodic points critical points Closure. A curved obtained by the Sym formula is smoothly closed of length L iff Λ 0 is a real critical point and a double point of σ d.

10 Finite-gap potentials and filaments Finite-gap potentials q have 2g + 2 simple points λ j σ d and g + 1 critical points α k. They can be written in terms of the Riemann θ-function of hyperelliptic Riemann surface Σ of genus g, with branch points λ j iex+int θ(ivx + iwt + r) q(x, t) = Ae, θ(ivx + iwt) together with their associated Baker-Akheizer eigenfunctions: φ(x, t; P) = eiω 1(P)x+iΩ 2 (P)t θ(ivx + iwt) [ e ( i E 2 x+i N 2 t) θ(ivx + iwt + r + θ 0 ) e (i E 2 x i N 2 t+ω 3(P)) θ(ivx + iwt + θ 0 ) A, E, N are real constants, the Ω i s are meromorphic differentials, and vectors V, W R g are determined by period integrals on Σ. ].

11 Isoperiodic Deformations Deforming spectral data while maintaining periodicity and closure Let λ 1,..., λ 2g+2 be branch points, and α 0,..., α g the critical points for a finite-gap NLS potential. For arbitrary real controls c 0 (ξ),..., c g (ξ), the deformation dλ j g dξ = k=0 c k λ j α k dα k dξ = l k c k + c l 1 2g+2 α l α k 2 j=1 c k λ j α k keeps frequencies V 1,..., V g constant (Grinevich & Schmidt 1995). Closure. If the Sym-Pohlmeyer formula at Λ 0 = α k produces a closed curve, then deformations with c k = 0 preserve closure.

12 Unpinching Starting in genus g = 0 with a plane wave potential (corresponding to a circular filament) i!! a 2!" plane we perform a sequence of closure-preserving isoperiodic deformations, each opening up a double point δ to two simple points λ s and a critical point α, thus increasing the genus by 1 at each step. a!i!! 2 Genus unpinches from 0 to 1

13 Deformation Schemes The notation [n; m 1,..., m g ], m k > n and gcd(n, m 1,..., m g ) = 1, describes a sequence of deformations that: begins with the n-times covered circle, with α 0 = 0 and selected double points located at δ k = sign(m k ) (m k /n) 2 1 opens up δ 1 = sign(m 1 ) (m 1 /n) 2 1, then opens up δ 2 sign(m 2 ) (m 2 /n) 2 1, and so on... Example: Deformation [4; 6, 13] (2,-3) torus knot (2,-13) cable knot

14 Cabling Theorem Each deformation step is a cabling operation Theorem: The scheme [n; m 1,..., m g ] produces a sequence of filaments γ (k), beginning with the unit circle, such that γ (k) is closed of length 2nπ/l k, where At any time t, γ (k) is a l k = gcd(n, m 1,..., m k ), 0 k g. ( lk 1 l k, m ) k cable about γ (k 1), l k Remark: Because m k > n, not all iterated torus knots are obtainable this way; e.g., can t use [4; 10, 3] to get a (2,3) cable on a (2,5).

15 Invariance of Cable Knot Type during the Evolution Still frames of the evolution of a (2,5)-cable on a trefoil knot. The dark yellow curve is the modified trajectory of a single point on the curve.

16 Stability of small amplitude torus knots A (p, q)-torus knot, with p and q co-prime, wraps the torus p times in the longitudinal direction and q times in the meridian direction. (Ricca, 2005). A small-amplitude (p, q)-torus knot evolving under the VFE is stable under linear perturbations iff q > p.!"#!"( Complex double point!"%!"&!!"#!!"'!!"$!"$!"'!"#!!"&!!"% The spectrum of a small amplitude finite-gap trefoil a (2, ±3) torus knot has a complex double point ν. Complex double points are associated with linear instabilities!!!"(!!"#

17 Complex double points and linear instabilities Recall the form of the Baker-Akheizer eigenfunction φ(x, t; P) = e iω 1(P)x+iΩ 2 (P)t F(iVx + iwt), where F is a bounded function of x and t (periodic in x for appropriate choice of the frequency vector V and quasiperiodic in time). Given a complex double point ν in the Floquet spectrum of an NLS potential, let P ± be the two points on the Riemann surface Σ that project to ν under the canonical projection π : Σ C. In general Im(Ω 2 (P ± )) 0, and the corresponding eigenfunctions φ ±, together with squared eigenfunctions φ ± φ ± grow exponentially in time. It follows that Finite-gap (p, q)-torus knots with q > p, close enough to a multiply covered circle, are unstable under linear perturbations of the VFE.

18 Visualizing linear instabilities: Bäcklund transformations Bäcklund transformations of integrable PDE s can be used to construct homoclinic orbits of unstable solutions. Gauge form of the NLS Bäcklund transformation: For φ +, φ two independent solutions of the AKNS system at (q, ν), define φ = c + φ + + c φ, c ± R, and construct the gauge matrix Then, G = N ( λ ν 0 0 λ ν ), with N = ( φ1 φ 2 φ1 φ 2 φ (1) (x, t, λ; ν) = G(λ; ν, φ)φ(x, t, λ) solves the AKNS system at (Q (1), λ), where Q (1) φ 1 φ2 = q + 2(ν ν) φ φ 2 2 is the corresponding new solution of the NLS equation. ). The new filament γ (1) is obtained from the Sym formula for φ (1).

19 Bäcklund transformation of the trefoil knot (time evolution) t0 t1 t2 t3 t4 t5 t6 t7 t8

20 Bäcklund transformation of (3, 5)-torus knot (t-evolution) t0 t1 t2 t t4 t5 t6 t t8 1 t t10 t11 1