Fractal solutions of dispersive PDE

Fractal solutions of dispersive PDE Burak Erdoğan (UIUC) ICM 2014 Satellite conference in harmonic analysis Chosun University, Gwangju, Korea, 08/05/14 In collaboration with V. Chousionis (U. Helsinki) and N. Tzirakis (UIUC). Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 1 / 30

V. Chousionis, M.B. Erdogan, and N. Tzirakis, Fractal solutions of linear and nonlinear dispersive partial differential equations, arxiv:1406.3283. M.B. Erdogan, N. Tzirakis, Talbot effect for the cubic nonlinear Schrodinger equation on the torus, to appear in Math. Res. Lett., arxiv:1303.3604. M.B. Erdogan, N. Tzirakis, Global smoothing for the periodic KdV evolution, Int. Math. Res. Notices (2013), 4589 4614. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 2 / 30

Linear Schrödinger on T: iu t + u xx = 0. Nonlinear Schrödinger on T: iu t + u xx + u 2 u = 0. KdV on T: u t + u xxx + uu x = 0. Vortex filament equation: u t = u x u xx, u : R T R 3. Schrödinger map: u t = u u xx, u : R T S 2. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 3 / 30

H. F. Talbot effect, 1836 Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 4 / 30

Berry et al, starting from 80s, studied this phenomenon using Linear Schrödinger on T Observations/conjectures: iu t + u xx = 0, u(0, x) = g(x) L 2 (T). u(t, x) = e it xx g = e itk 2 ĝ(k)e ikx. k= Quantization: At t = 2π p q, the solution is a linear combination of up to q translates of the initial data. For step function initial data, at t = 2πr, r irrational, the solution (real and imaginary parts) is a continuous curve with fractal dimension 3 2. The fractal dimensions of u(t, x) 2 and u(t, x) are also 3 2. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 5 / 30

The fractal dimension is 3 2 even there is a nonlinear perturbation. Fractal dimension (also known as Upper Minkowski dimension), dim(e), of a bounded set E: lim sup ɛ 0 log(n (E, ɛ)) log( 1 ɛ ), where N (E, ɛ) is the minimum number of ɛ balls required to cover E. Talbot effect for evolution equations: dependence of the qualitative behavior of the solution on the algebraic properties of time. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 6 / 30

Talbot effect for linear Schrödinger on the torus Evolved data at the given time for the initial data χ [0,π]. Figure from Chen and Olver, Dispersion of discontinuous periodic waves, preprint 2012, http://www.math.umn.edu/ olver Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 7 / 30

Berry and Klein (Also M. Taylor): For t = 2π p q To see this, note that e 2πi p q xx δ = 1 2π = 1 q q 1 e it xx g = 1 G p,q (j)g ( x 2π j ). q q k= q 1 j=0 e 2πik 2 p q e ikx = 1 q 1 e 2πil2 p q e ilx 2π l=0 = e 2πil2 p q e ilx 1 δ ( x 2π j ) q q l=0 q 1 [ q 1 j=0 l=0 q 1 j=0 ] e 2πil2 p q e 2πil j q δ ( x 2π j ). q j= e iqjx Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 8 / 30

Oskolkov (92): If g is of bounded variation, then e it xx g is continuous t in x if 2π Q. For t/2π Q, it is a bounded function with at most countably many discontinuities. Moreover, if g is also continuous then e it xx g Ct 0C0 x. Kapitanski-Rodniaski (99): e it xx g has better regularity properties (measured in Besov spaces) at irrational than rational times. In particular, for a.e. t, e it xx g C α, α < 1 2. Rodnianski (00): Assume that g BV (T) and g / ɛ>0 H 1 2 +ɛ (T). Then, for a.e. t, the graph of real or imaginary parts of e it xx g is a fractal curve with upper Minkowski dimension 3 2. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 9 / 30

Talbot effect for NLS on the torus Evolved data for NLS on [ π, π] with initial data χ [0,π]. Figure from Chen and Olver, Dispersion of discontinuous periodic waves, preprint 2012, http://www.math.umn.edu/ olver Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 10 / 30

Theorem (E., Tzirakis (13)) Consider the nonlinear Schrödinger equation on the torus t 2π t 2π iu t + u xx + u 2 u = 0, u(0, x) = g(x) BV. Q = u(t, x) is continuous in x, Q = u(t, x) is bounded with at most countably many discontinuities. If, in addition, g / ɛ>0 H 1 2 +ɛ (T). Then, for almost all times the graph of real or imaginary parts of u(t, ) has upper Minkowski dimension 3 2. Theorem (E., Tzirakis (13)) Fix s > 0. Assume that g H s (T). Then for any a < min(2s, 1/2), where P = g 2 2 /π. u(x, t) e i( xx +P)t g C 0 t R Hs+a x T, Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 11 / 30

NLS is globally wellposed in L 2 (T) (Bourgain) and illposed in H s for s < 0. Scaling: H 1 2 or Fl Earlier smoothing results: On R n setting: Bourgain (98), 2d cubic NLS (c.f. Keraani-Vargas (09)) There are many other results in unbounded domains. Christ (04): Fl p s Fl q s type (local) smoothing for cubic NLS on T. E., Tzirakis: KdV on T (11), Zakharov system on T (12). Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 12 / 30

Let R(u) = ( u 2 1 π u 2 L 2 ) u, and u X s,b = û(τ, k) k s τ k 2 b L 2, τ l 2 k For fixed s > 0 and a < min(2s, 1 2 ), we have R(u) X s+a,b 1 u 3 X s,b, provided that 0 < b 1 2 is sufficiently small. Corresponding bilinear estimates do not hold. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 13 / 30

Graph of e it xx g and e it xx g 2 : Theorem (Chousionis, E., Tzirakis, 2014) Let g be a nonconstant complex valued step function with jumps only at rational multiples of π. Then for a.e t, the graphs of both e it xx g and e it xx g 2 have fractal dimension 3 2. Main ingredient: for every irrational value of t/2π, e it xx g 2 H 1/2. Follows from the uniform distribution of {(kt, k 2 t) : k N} on T 2. Smoothing theorem above is not enough to extend this result to the NLS evolution. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 14 / 30

Fractal solutions for smoother data: Proposition. (Chousionis, E., Tzirakis, 2014) Let g : T C be of bounded variation and g ɛ>0 Hr0+ɛ for some r 0 [ 1 2, 3 4 ), then for almost all t both the real part and the imaginary part of the graph of e it xx g have dimension D [ 5 2 2r 0, 3 2 ]. Same statement holds for the NLS evolution. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 15 / 30

Higher order linear dispersive PDE: iu t + ( i x ) k u = 0, t R, x T = R/2πZ, k 3 u(x, 0) = g(x) BV. Proposition. (Chousionis, E., Tzirakis, 2014) Let g : T C be of bounded variation. Then e it( i x )k g is a continuous function of x for almost every t. Moreover if in addition g ɛ>0 H 1 2 +ɛ, then for almost all t both the real part and the imaginary part of the graph of e it( i x )k g have dimension D [1 + 2 1 k, 2 2 1 k ]. The smoothing estimate in [E., Tzirakis 2013] for the nonlinearity in the KdV equation implies that for almost all t the graph of the solution of KdV has dimension D [ 5 4, 7 4 ]. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 16 / 30

Vortex Filaments Approximation of the dynamics of a vortex filament under the Euler equations: γ t = γ x γ xx = κb, x arclength parameter, t time. γ(t, ) : arclength parametrized closed curve in R 3 Jerrard Smets ( 11) and de la Hoz Vega ( 13) studied the evolution of VF in the case when the initial data is a planar regular polygon. Simulations for the rational and irrational times: Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 17 / 30

Vortex filaments at rational times (initial data = square) Figure from Jerrard and Smets, On the motion of a curve by its binormal curvature, preprint 2011, arxiv:1109.5483v1 Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 18 / 30

Stereographic projection of the unit tangent vector at an irrational time (initial data = equilateral triangle) Figure from de la Hoz and Vega, Vortex filament equation for a regular polygon, preprint 2013, arxiv:1304.5521v1 Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 19 / 30

Vortex filament equation: γ t = γ x γ xx. Unit tangent vector u = γ x S 2. Differentiating the vortex filament equation (VF) we get u t = u u xx, Schrödinger Map Equation (SM). Hashimoto transformation: q(x, t) = κ(x, t)e i x 0 τ(x,t)dx, κ: curvature. τ: torsion. q satisfies NLS: iq t + q xx A(t)q + q 2 q = 0, A(t) R. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 20 / 30

VF in H s+1 level corresponds to SM in H s and NLS in H s 1 levels. SM on T: Ding Wang (98), Chang Shatah Uhlenbeck (00), Nahmod Shatah Vega Zeng (07), Rodnianski Rubinstein Staffilani (09): GWP in H 2 (T) Simulations above corresponds to SM with step function initial data and to NLS with initial data a sum of dirac deltas. We have fractal solutions of NLS in H 1 2 or in H s level, s ( 1 2, 3 4 ). We obtain global wellposedness of SM in H s, s > 3 2 holonomy initiaal data using: for identity Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 21 / 30

Frame construction of Chang Shatah Uhlenbeck and Nahmod Shatah Vega Zeng: Let u 0 : T S 2 be smooth, mean-zero and identity holonomy. Pick a unit vector e 0 (0) T u0 (0)S 2, and for x T define e 0 (x) T u0 (x)s 2 by parallel transport along u 0. e 0 is 2π-periodic since u 0 is identity holonomy. For each x, {e 0 (x), u 0 (x) e 0 (x)} is an orthonormal basis for T u0 (x)s 2. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 22 / 30

In this frame x u 0 (x) = q 1,0 (x)e 0 (x) + q 2,0 (x) u 0 (x) e 0 (x). Both q 1,0 and q 2,0 are 2π-periodic. Set q 0 := q 1,0 + iq 2,0 : T C and evolve q 0 by where q : T R C. iq t + q xx + 1 2 q 2 q = 0, In the following, p and q are related by p = p 1 + ip 2 := i x q. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 23 / 30

Theorem Let u, e solve the system of ODE t u = p 1 e + p 2 u e (1) t e = p 1 u 1 2 q 2 u e (2) e(x, 0) = e 0 (x), u(x, 0) = u 0 (x). (3) Then we have u : T R S 2, e is 2π-periodic in x, and for each x, t, e(x, t) T u(x,t) S 2, e(x, t) = 1, and x u = q 1 e + q 2 u e (4) x e = q 1 u. (5) Moreover, u is the unique 2π-periodic smooth solution of the Schrödinger map equation with initial data u(x, 0) = u 0 (x). Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 24 / 30

Following Chang-Shatah-Uhlenbeck, Nahmod-Shatah-Vega-Zeng we obtain: Global strong solutions of the SM for u(0, ) H s (T), s > 3 2, and identity holonomy. Talbot effect gives fractal curves that evolve for smoother initial curves; u(0, ) H s (T), s ( 3 2, 7 4 ). In particular the components of the curvature vector u x in the frame are fractal curves. Unique weak solutions of SM whose curvature is given by the NLS evolution. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 25 / 30

Proof of Oskolkov Kapitanski Rodnianski theorems for a. e. t For g BV, write ĝ(n) = 1 2π T e iny g(y)dy = 1 e iny dg(y) 2πin T e it xx g = ĝ(0) + lim N H N,t dg, where H N,t (x) = 0< n N e itn2 +inx n = 1 N 1 N T T n,t (x) N,t(x) + n(n + 1), n=1 where N [ T N,t (x) = e itn 2 +inx e itn2 inx ] n=1 Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 26 / 30

Well-known results from number theory: Weyl: Let t satisfy then t 2π a 1 q q 2, (6) sup T N,t (x) N + N log q + q log q. x q Khinchin, Levy (30 s): For a.e. t, ɛ > 0, and for all sufficiently large N, q [N, N 1+ɛ ] so that (6) holds for q. Combining these, for a. e. t, we have sup T N,t (x) N 1 2 +ɛ, ɛ > 0. x Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 27 / 30

For a. e. t, we have Recalling sup T N,t (x) N 1 2 +ɛ, ɛ > 0. x H N,t (x) = 1 N 1 N T T n,t (x) N,t(x) + n(n + 1), n=1 we see that, H N,t converges uniformly to a continuous function: H t (x) = n 0 e itn2 +inx This implies Oskolkov s theorem for a.e. t. n. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 28 / 30

This also implies that for a.e. t e it xx g ɛ>0 B 1 2 ɛ, (T), where the Besov space B s p, is defined via the norm: f B s p, := sup 2 sj P j f L p. j 0 This implies the upper bound for the dimension since C α (T) coincides with B α, and the fact that if f : T R is in C α, then the graph of f has upper Minkowski dimension D 2 α. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 29 / 30

If in addition we assume that g H r (T) for any r > 1/2. Then e it xx g ɛ>0 B 1 2 +ɛ 1, (T). Lower bound for the upper Minkowski dimension follows from Theorem (Deliu and Jawerth) The graph of a continuous function f : T R has upper Minkowski dimension D 2 s provided that f ɛ>0 Bs+ɛ 1,. Erdoğan (UIUC) Fractal solutions of dispersive PDE 08/05/14 30 / 30