Bielefeld Course on Nonlinear Waves - June 29, Department of Mathematics University of North Carolina, Chapel Hill. Solitons on Manifolds

Transcription

1 Joint work (on various projects) with Pierre Albin (UIUC), Hans Christianson (UNC), Jason Metcalfe (UNC), Michael Taylor (UNC), Laurent Thomann (Nantes) Department of Mathematics University of North Carolina, Chapel Hill Bielefeld Course on Waves - June 29, 2012

2 Outline

3 Ideas... We see a very nice trichotomy: H d : dispersion is so strong that only local nonlinearity dominates ([Christianson-M 2010, Mancini-Sandeep 2008]), R d : balance of dispersion and nonlinearity globally ([Strauss 1977,...]), (M, g) : Locally geometry dominates over nonlinearity ([Albin-Christianson-M-Thomann 2011]). See also work using Mountain Pass Theorems by several authors, for instance [del Piño, Pistoia,...]. Recent progress allows us to further explore the existence of global bound states on Weakly Homogeneous : [Christianson-M-Metcalfe-Taylor (2012)].

4 Ideas... We see a very nice trichotomy: H d : dispersion is so strong that only local nonlinearity dominates ([Christianson-M 2010, Mancini-Sandeep 2008]), R d : balance of dispersion and nonlinearity globally ([Strauss 1977,...]), (M, g) : Locally geometry dominates over nonlinearity ([Albin-Christianson-M-Thomann 2011]). See also work using Mountain Pass Theorems by several authors, for instance [del Piño, Pistoia,...]. Recent progress allows us to further explore the existence of global bound states on Weakly Homogeneous : [Christianson-M-Metcalfe-Taylor (2012)].

5 Optics Figure: Some "Gaussian Beams" in the work on nonlinear optics in the group of Ulf Peschel at MPI Science of Light.

6 The Technical Formulation We wish to explore existence of stationary solutions to the nonlinear Schrödinger equation on a manifold (M, g). Let g be the Laplace-Beltrami operator on M with respect to the metric g. Consider the nonlinear Schrödinger equation (NLS g) on M: { iu t + g u + u p u = 0, x M u(0, x) = u 0 (x). A state is a choice of initial condition R λ (x) such that u(t, x) = e iλt R λ (x) satisfies (NLS g) with initial data u(0, x) = R λ (x).

7 The Technical Formulation We wish to explore existence of stationary solutions to the nonlinear Schrödinger equation on a manifold (M, g). Let g be the Laplace-Beltrami operator on M with respect to the metric g. Consider the nonlinear Schrödinger equation (NLS g) on M: { iu t + g u + u p u = 0, x M u(0, x) = u 0 (x). A state is a choice of initial condition R λ (x) such that u(t, x) = e iλt R λ (x) satisfies (NLS g) with initial data u(0, x) = R λ (x).

8 The Technical Formulation Plugging in the ansatz yields the following stationary elliptic equation for R λ : g R λ + λr λ R λ p R λ = 0.

9 Existence in R d Existence of solitary waves for a wide variety of nonlinearities is proved in Berestycki-Lions by minimizing the quantity T (u) = u 2 dx R d with respect to the constraint V (u) := λ2 2 u 2 dx + F( u )dx = 1. R d R d

10 Existence Then, using a minimizing sequence and Schwarz symmetrization, one sees the existence of the nonnegative, spherically symmetric, decreasing soliton solution. For uniqueness, see McLeod, where a shooting method is implemented to show that the desired soliton behavior only occurs for one particular initial value.

11 Existence Then, using a minimizing sequence and Schwarz symmetrization, one sees the existence of the nonnegative, spherically symmetric, decreasing soliton solution. For uniqueness, see McLeod, where a shooting method is implemented to show that the desired soliton behavior only occurs for one particular initial value.

12 From the work of P.L. Lions, we have the following means of finding constrained minimizers: Let (ρ n ) n 1 be a sequence in L 1 (R d ) satisfying: ρ n 0 in R d, R d ρ n dx = λ where λ > 0 is fixed. Then there exists a subsequence (ρ nk ) k 1 satisfying one of the three following possibilities:

13 From the work of P.L. Lions, we have the following means of finding constrained minimizers: Let (ρ n ) n 1 be a sequence in L 1 (R d ) satisfying: ρ n 0 in R d, R d ρ n dx = λ where λ > 0 is fixed. Then there exists a subsequence (ρ nk ) k 1 satisfying one of the three following possibilities:

14 i. (compactness) there exists y k R d such that ρ nk ( + y nk ) is tight, i.e.: ɛ > 0, R <, ρ nk (x)dx λ ɛ; y k +B R ii. (vanishing) lim k sup y R d y+b R ρ nk (x)dx = 0, for all R < ;

15 i. (compactness) there exists y k R d such that ρ nk ( + y nk ) is tight, i.e.: ɛ > 0, R <, ρ nk (x)dx λ ɛ; y k +B R ii. (vanishing) lim k sup y R d y+b R ρ nk (x)dx = 0, for all R < ;

16 iii. (dichotomy) there exists α [0, λ] such that for all ɛ > 0, there exists k 0 1 and ρ 1 k, ρ2 k L1 +(R d ) satisfying for k k 0 : ρ nk (ρ 1 k + ρ2 k ) L 1 ɛ, R ρ 1 d kdx α ɛ, R ρ 2 d kdx (λ α) ɛ, d(supp(ρ 1 k ), Supp(ρ2 k )).

17 Existence The idea for H d from [Christianson-M 2010] relies on conjugating H d into an operator on Euclidean space, and then finding minimizers for the energy functional. See also [Mancini-Sandeep 2008]. The problem of minimizing the functional is greatly simplified assuming the functions involved depend only on the radius r = x, as then the minimization theory in R d may be used. Let us define a space Hr 1 to be the space of all spherically symmetric functions in H 1.

18 Existence The idea for H d from [Christianson-M 2010] relies on conjugating H d into an operator on Euclidean space, and then finding minimizers for the energy functional. See also [Mancini-Sandeep 2008]. The problem of minimizing the functional is greatly simplified assuming the functions involved depend only on the radius r = x, as then the minimization theory in R d may be used. Let us define a space Hr 1 to be the space of all spherically symmetric functions in H 1.

19 Existence The idea for H d from [Christianson-M 2010] relies on conjugating H d into an operator on Euclidean space, and then finding minimizers for the energy functional. See also [Mancini-Sandeep 2008]. The problem of minimizing the functional is greatly simplified assuming the functions involved depend only on the radius r = x, as then the minimization theory in R d may be used. Let us define a space Hr 1 to be the space of all spherically symmetric functions in H 1.

20 Radial Symmetry Lemma Let u be a solution to Hyperbolic NLS with initial data u 0 Hr 1 and the nonlinearity f ( u )u = u p u with 4 d 2 > p > 0. Then u H1 r. The proof of this lemma is by uniqueness, which follows from the implicit local uniqueness following from the Strichartz estimates in Ionescu-Staffilani.

21 Radial Symmetry Lemma Let u be a solution to Hyperbolic NLS with initial data u 0 Hr 1 and the nonlinearity f ( u )u = u p u with 4 d 2 > p > 0. Then u H1 r. The proof of this lemma is by uniqueness, which follows from the implicit local uniqueness following from the Strichartz estimates in Ionescu-Staffilani.

22 Radial Symmetry We show that any constrained minimizer in hyperbolic space may be replaced by one that is spherically symmetric, so that we may neglect the angular derivative. To do this, we modify the standard argument of Lieb-Loss in R d, using heat kernel arguments to show symmetric decreasing rearrangement or Schwarz symmetrization lowers the kinetic energy in H d. The symmetric decreasing rearrangement on H d is given by f (Ω) = inf{t : λ f (t) µ(b(dist (Ω, 0)))}, where µ is the natural measure on H d, dist is the hyperbolic distance function on H d and λ f (t) = µ({ f > t}).

23 Radial Symmetry We show that any constrained minimizer in hyperbolic space may be replaced by one that is spherically symmetric, so that we may neglect the angular derivative. To do this, we modify the standard argument of Lieb-Loss in R d, using heat kernel arguments to show symmetric decreasing rearrangement or Schwarz symmetrization lowers the kinetic energy in H d. The symmetric decreasing rearrangement on H d is given by f (Ω) = inf{t : λ f (t) µ(b(dist (Ω, 0)))}, where µ is the natural measure on H d, dist is the hyperbolic distance function on H d and λ f (t) = µ({ f > t}).

24 Radial Symmetry We show that any constrained minimizer in hyperbolic space may be replaced by one that is spherically symmetric, so that we may neglect the angular derivative. To do this, we modify the standard argument of Lieb-Loss in R d, using heat kernel arguments to show symmetric decreasing rearrangement or Schwarz symmetrization lowers the kinetic energy in H d. The symmetric decreasing rearrangement on H d is given by f (Ω) = inf{t : λ f (t) µ(b(dist (Ω, 0)))}, where µ is the natural measure on H d, dist is the hyperbolic distance function on H d and λ f (t) = µ({ f > t}).

25 Radial Symmetry First of all, it is clear f is spherically symmetric, nonincreasing, lower semicontinuous and for any 1 p. f L p (H d ) = f L p (H d )

26 Radial Symmetry Lemma Suppose f H 1 (H d ), and f is the symmetric decreasing rearrangement of f. Then f L 2 (H d ) f L 2 (H d ).

27 Radial Symmetry We use standard Hilbert space theory as in Lieb-Loss. Namely, we observe that the kinetic energy satisfies where f L 2 (H d ) = lim t 0 It (f ), I t (f ) = t 1 [(f, f ) H d (f, e H d t f ) H d ] and (, ) H d is the natural L 2 inner-product on H d.

28 Radial Symmetry As (f, f ) = (f, f ) by construction, we need (f, e H d t f ) H d (f, e H d t f ) H d in order to see that symmetrization decreases the kinetic energy.

29 Radial Symmetry In R d, this is done using convolution operators and the Riesz rearrangement inequality, which we do not have here. Instead, we use heat kernel decay and an a rearrangement theorem of Draghici.

30 Radial Symmetry For each t > 0, the heat kernel on hyperbolic space, p d (ρ, t), is a decreasing function of the hyperbolic distance ρ.

31 Radial Symmetry From Draghici, we have used the following theorem: Theorem (Draghici) Let X = H d, f i : X R + be m nonnegative functions, Ψ AL 2 (R m +) be continuous and K ij : [0, ) [0, ), i < j, j {1,..., m} be decreasing functions. We define I[f 1,..., f m ] = Ψ(f 1 (Ω 1 ),..., f m (Ω m )) X m Π i<j K ij (d(ω i, Ω j ))dω 1... dω m. Then, the following inequality holds: I[f 1,..., f m ] I[f 1,..., f m].

32 Existence In this section we begin to analyze HNLS in the case f ( u )u is a so-called focusing nonlinearity. From the polar form of H d, we approach the problem by comparison to the standard Laplacian on R d. In this direction, let us recall that the metric for R d in polar coordinates is given by ds 2 = dr 2 + r 2 dω 2, so that the Jacobian is r d 1.

33 Existence In this section we begin to analyze HNLS in the case f ( u )u is a so-called focusing nonlinearity. From the polar form of H d, we approach the problem by comparison to the standard Laplacian on R d. In this direction, let us recall that the metric for R d in polar coordinates is given by ds 2 = dr 2 + r 2 dω 2, so that the Jacobian is r d 1.

34 Existence Similarly, the Jacobian from the polar coordinate representation of H d is sinh d 1 r. We employ an isometry T taking L 2 (r d 1 drdω) to L 2 (sinh d 1 rdrdω), so that T 1 ( H d )T is a non-negative, unbounded, essentially self-adjoint operator on L 2 (R d ).

35 Existence We define and take Tu = φu. ( r φ(r) = sinh r ( sinh r φ 1 (r) = r ) d 1 2 ) d 1 2 Conjugating H d by φ, we have a second order differential operator on R d with the leading order term almost the Laplacian on R d.,,

36 Existence We define and take Tu = φu. ( r φ(r) = sinh r ( sinh r φ 1 (r) = r ) d 1 2 ) d 1 2 Conjugating H d by φ, we have a second order differential operator on R d with the leading order term almost the Laplacian on R d.,,

37 Existence Indeed, we have [ ( ) ] φ 1 ( H d )(φu) = u d V d (r) + u. 2

38 Existence Here so that V 0 (r) = 1 d, r (d 1)(d 3) V d (r) = 4 ( ) r 2 sinh 2 r r 2 sinh 2, r = R d r 2 sinh 2 r r 2 sinh 2 S d 1. r

39 Existence For completeness, we record the following simple lemma. Lemma The function Ṽ = sinh2 r r 2 r 2 sinh 2 r satisfies the following properties: (i) Ṽ C (R), (ii) Ṽ 0, (iii) Ṽ (0) = 1 3, (iv) Ṽ = O(r 2 ), r, and (v) Ṽ (r) = 0 only at r = 0.

40 Existence Remark Note that the potential V 3 = 0, and the lemma implies V 2 0 has a bump at 0, while for d 4, the potential V d 0 has a well at 0.

41 Existence After this conjugation to R d, Hyperbolic NLS becomes { iu t u + (d 1)2 4 u + V d (x)u f (x, u) = 0, x R d u(0, x) = u 0 (x) H 1, where now the nonlinearity f takes the following form after conjugation: f (x, u) = φ 1 f (φu)(φu) = ( r ) p(d 1) 2 u p u. sinh r

42 Existence We have the naturally defined conserved quantities Q(u) = u 2 L 2 and E(u) = R d [ 1 2 u a(x) angu 2 (d 1)2 (V d ( x ) + 4 ) ] u 2 F(x, u) dx.

43 Existence Here a(x) = x 2 sinh 2 x x 2 sinh 2 x is the offset of the spherical Laplacian in the definition of, F(x, u) = = u(x) 0 1 p + 2 f (x, s)ds =: K ( x ) u p+2. ( ) x (d 1)p/2 u p+2 sinh x

44 Existence From the work of Banica, we have global existence for p < 4 d and finite time blow-up for 4 d p < 4 d 2.

45 Existence We make a soliton ansatz for Hyperbolic NLS in R d : u(x, t) = e iλt R λ, for a function R λ depending on a real parameter (the soliton parameter) λ > 0. Plugging this ansatz into the conjugated equation Hyperbolic NLS we see we must have ( ) (d 1) 2 R λ + + λ + V d (r) R λ 4 f (x, R λ ) = 0. Hence, we seek a minimizer of the associated energy functional to this nonlinear elliptic equation for u L 2 fixed.

46 Existence We make a soliton ansatz for Hyperbolic NLS in R d : u(x, t) = e iλt R λ, for a function R λ depending on a real parameter (the soliton parameter) λ > 0. Plugging this ansatz into the conjugated equation Hyperbolic NLS we see we must have ( ) (d 1) 2 R λ + + λ + V d (r) R λ 4 f (x, R λ ) = 0. Hence, we seek a minimizer of the associated energy functional to this nonlinear elliptic equation for u L 2 fixed.

47 Existence We make a soliton ansatz for Hyperbolic NLS in R d : u(x, t) = e iλt R λ, for a function R λ depending on a real parameter (the soliton parameter) λ > 0. Plugging this ansatz into the conjugated equation Hyperbolic NLS we see we must have ( ) (d 1) 2 R λ + + λ + V d (r) R λ 4 f (x, R λ ) = 0. Hence, we seek a minimizer of the associated energy functional to this nonlinear elliptic equation for u L 2 fixed.

48 Existence We note that the continuous spectrum is shifted according to the term ( ) d 1 2 u. 2 In the end, this term does not alter the existence argument for soliton solutions, however, it does expand the allowed range of soliton parameters from λ (0, ) to ( ) d 1 2 λ (, ). 2

49 Existence We note that the continuous spectrum is shifted according to the term ( ) d 1 2 u. 2 In the end, this term does not alter the existence argument for soliton solutions, however, it does expand the allowed range of soliton parameters from λ (0, ) to ( ) d 1 2 λ (, ). 2

50 Existence Hence, we set ( ) d 1 2 µ d = λ + > 0. 2

51 Quasimodes By separating variables in the t direction, we write ψ(x, t) = e iλt u(x), from which we get the stationary equation (λ g )u = σ u p u. The construction in the proof finds a function u λ (x) = λ (d 1)/8 g(λ 1/4 x) such that g is rapidly decaying away from Γ, C, g is normalized in L 2, and (λ g )u λ = σ u λ p u λ + E(u λ ), where the error E(u λ ) is expressed by the a truncation of an asymptotic series similar to that in the work of Thomann and is of lower order in λ.

52 Quasimodes By separating variables in the t direction, we write ψ(x, t) = e iλt u(x), from which we get the stationary equation (λ g )u = σ u p u. The construction in the proof finds a function u λ (x) = λ (d 1)/8 g(λ 1/4 x) such that g is rapidly decaying away from Γ, C, g is normalized in L 2, and (λ g )u λ = σ u λ p u λ + E(u λ ), where the error E(u λ ) is expressed by the a truncation of an asymptotic series similar to that in the work of Thomann and is of lower order in λ.

53 Quasimodes The result is an improvement over the trivial approximate solution. It is well known that there exist quasimodes for the linear equation localized near Γ of the form v λ (x) = λ (d 1)/8 e isλ1/2 f (s, λ 1/4 x), (λ > 0), with f a function rapidly decaying away from Γ, and s a parametrization around Γ, so that v λ (x) satisfies (λ g )v λ = O(λ ) v λ in any seminorm, see Ralston.

54 Quasimodes Then (λ g )v λ = σ v λ p v λ + E 2 (v λ ), where the error E 2 (v λ ) = v λ p v λ satisfies E 2 (v λ ) Ḣs = O(λ s/2+p(d 1)/8 ).

55 Toy Model In this section we consider a toy model in two dimensions. As it is a toy model, we will not dwell on error analysis, and instead make Taylor approximations at will without remarking on the error terms. Consider the manifold M = R x /2πZ R θ /2πZ, equipped with a metric of the form ds 2 = dx 2 + A 2 (x)dθ 2, where A C is a smooth function, A ɛ > 0 for some ɛ.

56 Toy Model From this metric, we get the volume form dvol = A(x)dxdθ, and the Laplace-Beltrami operator acting on 0-forms g f = ( x 2 + A 2 θ 2 + A 1 A x )f.

57 Quasimodes We observe that we can conjugate g by an isometry of metric spaces and separate variables so that spectral analysis of g is equivalent to a one-variable semiclassical problem with potential. Let S : L 2 (X, dvol) L 2 (X, dxdθ) be the isometry given by Su(x, θ) = A 1/2 (x)u(x, θ).

58 Quasimodes We observe that we can conjugate g by an isometry of metric spaces and separate variables so that spectral analysis of g is equivalent to a one-variable semiclassical problem with potential. Let S : L 2 (X, dvol) L 2 (X, dxdθ) be the isometry given by Su(x, θ) = A 1/2 (x)u(x, θ).

59 Quasimodes A simple calculation gives f = ( x 2 A 2 (x) θ 2 + V 1(x))f, where the potential V 1 (x) = 1 2 A A (A ) 2 A 2.

60 Existence We are interested in the nonlinear Schrödinger equation, so we make a separated ansatz: u λ (t, x, θ) = e itλ e ikθ ψ(x), where k Z and ψ is to be determined (depending on both λ and k).

61 Existence Applying the Schrödinger operator (with replacing ) to u λ yields the equation with (D t + )e itλ e ikθ ψ(x) = σ ψ p e itλ e ikθ ψ(x) 2 = 2 d 1 p < 4 d 1 = 4, where we have used the standard notation D = i.

62 Existence We are interested in the behaviour of a solution or approximate solution near an elliptic periodic geodesic, which occurs at a maximum of the function A. For simplicity, let A(x) = (1 + cos 2 (x))/2, so that in a neighbourhood of x = 0, A 2 1 x 2 and A x 2.

63 Existence We are interested in the behaviour of a solution or approximate solution near an elliptic periodic geodesic, which occurs at a maximum of the function A. For simplicity, let A(x) = (1 + cos 2 (x))/2, so that in a neighbourhood of x = 0, A 2 1 x 2 and A x 2.

64 Existence The function V 1 (x) const. in a neighbourhood of x = 0, so we will neglect V 1. If we assume ψ(x) is localized near x = 0, we get the stationary reduced equation ( λ + 2 x k 2 (1 + x 2 ))ψ = σ ψ p ψ.

65 Existence Let h = k 1 and use the rescaling operator T ψ(x) = T h,0 ψ(x) = h 1/4 ψ(h 1/2 x) ( below with n = 1) to conjugate: or T 1 ( λ + 2 x k 2 (1 + x 2 ))TT 1 ψ = T 1 (σ ψ p ψ) ( λ + h 1 2 x k 2 (1 + hx 2 )φ = σh p/4 φ p φ, where φ = T 1 ψ.

66 Existence Let us now multiply by h: where ( 2 + x 2 E)φ = σh q φ p φ, E = 1 λh2 h and q = 1 p d 1 = 1 p 4 4. Observe the range restriction on p is precisely so that 0 < q 1/2.

67 Existence We make a WKB type ansatz, although in practice we will only take two terms (more is possible if the nonlinearity is smooth): φ = φ 0 + h q φ 1, E = E 0 + h q E 1. The first two equations are h 0 : ( 2 + x 2 E 0 )φ 0 = 0, h q : ( 2 + x 2 E 0 h q E 1 )φ 1 = E 1 φ 0 + σ φ 0 p φ 0. Observe we have included the h q E 1 φ 1 term on the left hand side.

68 Existence We make a WKB type ansatz, although in practice we will only take two terms (more is possible if the nonlinearity is smooth): φ = φ 0 + h q φ 1, E = E 0 + h q E 1. The first two equations are h 0 : ( 2 + x 2 E 0 )φ 0 = 0, h q : ( 2 + x 2 E 0 h q E 1 )φ 1 = E 1 φ 0 + σ φ 0 p φ 0. Observe we have included the h q E 1 φ 1 term on the left hand side.

69 Applications General theory of states on symmetric spaces - What carries over from Euclidean Space?? Spectral Properties?? Uniqueness?? Sharp inequality bounds?? Stability Theory - Related to spectral properties of the linearized operator, uniqueness,... Traveling Waves - Motion generate by symmetries of the manifold and killing vector fields that commute with the Laplacian... lots of symmetry required.

70 Applications General theory of states on symmetric spaces - What carries over from Euclidean Space?? Spectral Properties?? Uniqueness?? Sharp inequality bounds?? Stability Theory - Related to spectral properties of the linearized operator, uniqueness,... Traveling Waves - Motion generate by symmetries of the manifold and killing vector fields that commute with the Laplacian... lots of symmetry required.

71 Applications General theory of states on symmetric spaces - What carries over from Euclidean Space?? Spectral Properties?? Uniqueness?? Sharp inequality bounds?? Stability Theory - Related to spectral properties of the linearized operator, uniqueness,... Traveling Waves - Motion generate by symmetries of the manifold and killing vector fields that commute with the Laplacian... lots of symmetry required.