Soliton-like Solutions to NLS on Compact Manifolds

Transcription

1 Soliton-like Solutions to NLS on Compact Manifolds H. Christianson joint works with J. Marzuola (UNC), and with P. Albin (Jussieu and UIUC), J. Marzuola, and L. Thomann (Nantes) Department of Mathematics University of North Carolina - Chapel Hill July 20, 2011

2 Outline 1 Introduction: Geometry and Bound States

3 Outline 1 Introduction: Geometry and Bound States 2

4 Goal of talk Goal: Understand interplay between local geometry, global geometry, and nonlinear PDEs.

5 Goal of talk Goal: Understand interplay between local geometry, global geometry, and nonlinear PDEs. For this talk, we examine the nonlinear effect of bound state (sometimes called soliton) formation.

6 Goal of talk Goal: Understand interplay between local geometry, global geometry, and nonlinear PDEs. For this talk, we examine the nonlinear effect of bound state (sometimes called soliton) formation. Global geometry: trapping, non-trapping, structure of infinity, global curvatures

7 Goal of talk Goal: Understand interplay between local geometry, global geometry, and nonlinear PDEs. For this talk, we examine the nonlinear effect of bound state (sometimes called soliton) formation. Global geometry: trapping, non-trapping, structure of infinity, global curvatures Local geometry: periodic geodesics, local sectional curvatures

8 Goal of talk Goal: Understand interplay between local geometry, global geometry, and nonlinear PDEs. For this talk, we examine the nonlinear effect of bound state (sometimes called soliton) formation. Global geometry: trapping, non-trapping, structure of infinity, global curvatures Local geometry: periodic geodesics, local sectional curvatures Goal: Three different scenarios where geometry plays a different role.

9 Geometric trichotomy There are three interesting regimes: In R d, dispersion and nonlinearity must balance for bound states to form

10 Geometric trichotomy There are three interesting regimes: In R d, dispersion and nonlinearity must balance for bound states to form In H d, dispersion is very strong at infinity, and nonlinearity acts (almost) locally

11 Geometric trichotomy There are three interesting regimes: In R d, dispersion and nonlinearity must balance for bound states to form In H d, dispersion is very strong at infinity, and nonlinearity acts (almost) locally If M is compact, local geometry wins over nonlinearity

12 Bound states in R d (Focusing) Nonlinear Schrödinger equation (NLS): { iu t + u + u p u = 0, in R R d u(0, x) = u 0 (x).

13 Bound states in R d (Focusing) Nonlinear Schrödinger equation (NLS): { iu t + u + u p u = 0, in R R d u(0, x) = u 0 (x). A nonlinear bound state is a choice of initial condition R λ (x) such that satisfies NLS u(t, x) = e iλt R λ (x)

14 Bound states in R d (Focusing) Nonlinear Schrödinger equation (NLS): { iu t + u + u p u = 0, in R R d u(0, x) = u 0 (x). A nonlinear bound state is a choice of initial condition R λ (x) such that satisfies NLS stationary equation: u(t, x) = e iλt R λ (x) R λ + λr λ R λ p R λ = 0.

15 Bound states in R d (Focusing) Nonlinear Schrödinger equation (NLS): { iu t + u + u p u = 0, in R R d u(0, x) = u 0 (x). A nonlinear bound state is a choice of initial condition R λ (x) such that satisfies NLS stationary equation: u(t, x) = e iλt R λ (x) R λ + λr λ R λ p R λ = 0. Try to solve using calculus of variations energy functional without a sign!

16 An existence theorem Theorem (Strauss 77, Berestycki-Lions 83) There exists such a bound state for every λ > 0 and every 0 < p < 4 d 2.

17 An existence theorem Theorem (Strauss 77, Berestycki-Lions 83) There exists such a bound state for every λ > 0 and every 0 < p < 4 d 2. Proof involves careful ruling out of mass escaping to, called concentration compactness shows a delicate balance between dispersion and nonlinearity focusing the mass.

18 Bound states in H d Consider H d -NLS: { iu t + H d u + u p u = 0, in R H d u(0, x) = u 0 (x).

19 Bound states in H d Consider H d -NLS: { iu t + H d u + u p u = 0, in R H d u(0, x) = u 0 (x). Theorem (C -Marzuola 10) There exists a bound state for λ > (d 1)2, 0 < p < 4 2 d 2.

20 Remarks Larger range of λ is due to spectrum of H d. Range of p is same as R d!

21 Remarks Larger range of λ is due to spectrum of H d. Range of p is same as R d! Proof shows mass cannot escape to, so the problem is morally a local one. Bypasses concentration compactness (mostly). E.g. the problem is conjugate to one with exponentially decaying weights.

22 Remarks Larger range of λ is due to spectrum of H d. Range of p is same as R d! Proof shows mass cannot escape to, so the problem is morally a local one. Bypasses concentration compactness (mostly). E.g. the problem is conjugate to one with exponentially decaying weights. Really exploits spherical symmetry!

23 Remarks Larger range of λ is due to spectrum of H d. Range of p is same as R d! Proof shows mass cannot escape to, so the problem is morally a local one. Bypasses concentration compactness (mostly). E.g. the problem is conjugate to one with exponentially decaying weights. Really exploits spherical symmetry! Work is in progress with J. Marzuola, J. Metcalfe, and M. Taylor to extend to more complicated geometries. Difficulty is fewer symmetries.

24 Linear concentration on compact manifolds Expect linear effects from local geometry to dominate over nonlinear effects, at least for small times.

25 Linear concentration on compact manifolds Expect linear effects from local geometry to dominate over nonlinear effects, at least for small times. Suppose (M, g) is compact, M =, γ M is a stable periodic geodesic.

26 Linear concentration on compact manifolds Expect linear effects from local geometry to dominate over nonlinear effects, at least for small times. Suppose (M, g) is compact, M =, γ M is a stable periodic geodesic. Theorem (Ralston 71,...) There is a sequence of quasimodes: { ( g λ)u = O(λ ), u L 2 (M) = 1, such that u L 2 (M\neigh (γ)) = O(λ ).

27 Linear concentration on compact manifolds Expect linear effects from local geometry to dominate over nonlinear effects, at least for small times. Suppose (M, g) is compact, M =, γ M is a stable periodic geodesic. Theorem (Ralston 71,...) There is a sequence of quasimodes: { ( g λ)u = O(λ ), u L 2 (M) = 1, such that u L 2 (M\neigh (γ)) = O(λ ). Does the same thing happen for the nonlinear stationary equation?

28 Examples Some examples: M = S d, M a Zoll manifold, M an ellipsoid, M lumpy torus:

29 A toy model: the lumpy torus Consider the lumpy torus: M = R x /2πZ R θ /2πZ, with metric and Laplacian g = dx 2 + A 2 (x)dθ 2, g f = ( x 2 + A 2 θ 2 )f + l.o.t.

30 A toy model: the lumpy torus Consider the lumpy torus: M = R x /2πZ R θ /2πZ, with metric and Laplacian g = dx 2 + A 2 (x)dθ 2, g f = ( x 2 + A 2 θ 2 )f + l.o.t. Take A(x) = (1 + cos 2 (x))/2, max at x = 0, A 2 1 x 2 and A x 2. = stable periodic geodesic at x = 0.

31 Ansatz Separated ansatz (based on Thomann 08): u λ (t, x,θ) = e itλ e ikθ ψ(x),

32 Ansatz Separated ansatz (based on Thomann 08): u λ (t, x,θ) = e itλ e ikθ ψ(x), (D t + )u λ = (λ+ 2 x k 2 A 2 (x))e itλ e ikθ ψ(x) = σ ψ p e itλ e ikθ ψ(x), up to l.o.t. Here σ = ±1 implies focusing or defocusing!

33 Ansatz Separated ansatz (based on Thomann 08): u λ (t, x,θ) = e itλ e ikθ ψ(x), (D t + )u λ = (λ+ 2 x k 2 A 2 (x))e itλ e ikθ ψ(x) = σ ψ p e itλ e ikθ ψ(x), up to l.o.t. Here σ = ±1 implies focusing or defocusing! Assume ψ(x) localized near x = 0, ( λ + 2 x k 2 (1 + x 2 ))ψ = σ ψ p ψ. Looks like harmonic oscillator on LHS!

34 Ansatz Separated ansatz (based on Thomann 08): u λ (t, x,θ) = e itλ e ikθ ψ(x), (D t + )u λ = (λ+ 2 x k 2 A 2 (x))e itλ e ikθ ψ(x) = σ ψ p e itλ e ikθ ψ(x), up to l.o.t. Here σ = ±1 implies focusing or defocusing! Assume ψ(x) localized near x = 0, ( λ + 2 x k 2 (1 + x 2 ))ψ = σ ψ p ψ. Looks like harmonic oscillator on LHS! Let h = k 1 and rescale: ϕ(x) = h 1/4 ψ(h 1/2 x) where ( 2 x + x 2 E)ϕ = σh 1 p/4 ϕ p ϕ, E := 1 λh2 h

35 WKB Assume 2 p < 4 so that if q = 1 p/4, 0 < q 1/2.

36 WKB Assume 2 p < 4 so that if q = 1 p/4, 0 < q 1/2. WKB ansatz: ϕ = ϕ 0 + h q ϕ 1, E = E 0 + h q E 1.

37 WKB Assume 2 p < 4 so that if q = 1 p/4, 0 < q 1/2. WKB ansatz: ϕ = ϕ 0 + h q ϕ 1, E = E 0 + h q E 1. h 0 : ( 2 x + x 2 E 0 )ϕ 0 = 0, h q : ( 2 x + x 2 E 0 h q E 1 )ϕ 1 = E 1 ϕ 0 + σ ϕ 0 p ϕ 0.

38 WKB Assume 2 p < 4 so that if q = 1 p/4, 0 < q 1/2. WKB ansatz: ϕ = ϕ 0 + h q ϕ 1, E = E 0 + h q E 1. h 0 : ( x 2 + x 2 E 0 )ϕ 0 = 0, h q : ( x 2 + x 2 E 0 h q E 1 )ϕ 1 = E 1 ϕ 0 + σ ϕ 0 p ϕ 0. The first equation is easy: ϕ 0 (x) = e x2 /2, E 0 = 1.

39 WKB Assume 2 p < 4 so that if q = 1 p/4, 0 < q 1/2. WKB ansatz: ϕ = ϕ 0 + h q ϕ 1, E = E 0 + h q E 1. h 0 : ( x 2 + x 2 E 0 )ϕ 0 = 0, h q : ( x 2 + x 2 E 0 h q E 1 )ϕ 1 = E 1 ϕ 0 + σ ϕ 0 p ϕ 0. The first equation is easy: ϕ 0 (x) = e x2 /2, E 0 = 1. For second equation, project away from ϕ 0 E 1 ϕ 0 + σ ϕ 0 p ϕ 0,ϕ 0 = 0, fixes E 1.

40 WKB ansatz II Rescale back (D t + )u = σ u p u + Q, where Q satisfies the pointwise bound Q = O(h q u p+1 ).

41 A general theorem In general, the technique works for high regularity with smooth nonlinearities: Theorem Let (M, g) be a compact Riemannian manifold of dimension d 2, γ M stable periodic geodesic. Let p be an even integer, let s 0 and assume that p ( d 1 4 s ) < 1. Then for h 1 sufficiently small, and any δ > 0, there exists ϕ h (x) H (M) satisfying (i) Frequency localization : For all r 0, ϕ h H r (M) h s r.

42 A general theorem II Theorem (continued...) (ii) Spatial localization near γ : ϕ h H s (M\neigh (γ) = O(h ) ϕ h H s (M). (iii) Nonlinear quasimode : With λ(h) = h 2 E 0 h , g ϕ h = λ(h)ϕ h σ ϕ h p ϕ h + O(h ).

43 Coherence in NLS Under the same assumptions, we estimate how long the approximate solution stays coherent: Theorem Let u h (t, x) solve IVP with u h (0, x) = ϕ h (x). Then u h L ([0,T h ];H s (M\neigh (γ))) Ch (d+1)/4. d 1 p( for T h = c 0 h 4 s) ln( 1 h ).

44 Remarks In other words, there is a highly concentrated solution along γ, oscillating at frequency h 1, which stays coherent at least for short times

45 Remarks In other words, there is a highly concentrated solution along γ, oscillating at frequency h 1, which stays coherent at least for short times Techniques work for non-smooth nonlinearities, but with worse error estimates

46 Remarks In other words, there is a highly concentrated solution along γ, oscillating at frequency h 1, which stays coherent at least for short times Techniques work for non-smooth nonlinearities, but with worse error estimates We think for analytic manifolds, the timescale is h 1 better (work in progress)