Université de Nantes 3rd Meeting of the GDR Quantum Dynamics MAPMO, Orléans, 2-4 February 2011. (Joint work with Benoît Grébert)
Introduction The equation : We consider the nonlinear Schrödinger equation with harmonic potential ( i tu + 2 x u x 2 u = u 2 u, (t, x) R R, u(0, x) = u 0(x), Physical interest : Model for Bose-Einstein condensates. Litterature : R. Fukuizumi, K. Yajima - G. Zang, R. Carles,... Motivation : The equation is globally well-posed in the energy space. Let p > 1. Behaviour of u(t) H p (R) when t?
Introduction Difficulty : Spectral structure of 2 x + x 2 : the eigenvalues are λ j = 2j 1, j 1 and are completely resonant in the sense that there exist many k N of finite length so that k λ = P j 1 k jλ j = 0. Therefore, we consider ( i tu + 2 x u x 2 u + εv (x)u = ε u 2 u, (t, x) R R, u(0, x) = u 0(x). (NLS) where ε 1 and V S(R, R).
Introduction Difficulty : Spectral structure of 2 x + x 2 : the eigenvalues are λ j = 2j 1, j 1 and are completely resonant in the sense that there exist many k N of finite length so that k λ = P j 1 k jλ j = 0. Therefore, we consider ( i tu + 2 x u x 2 u + εv (x)u = ε u 2 u, (t, x) R R, u(0, x) = u 0(x). (NLS) where ε 1 and V S(R, R). Aim : Construction of quasi-periodic in time solutions to (NLS) for typical V. In particular, the H p norm of these solutions will be bounded. Quasi-periodicity : f : R C, t f (t) is quasi-periodic if there exist n 1, a periodic function U : T n C and (ω 1,..., ω n) R n so that for all t R, f (t) = U(ω 1t,..., ω nt).
Introduction Denote by A = 2 x + x 2 εv (x). There exists an Hilbertian basis of L 2 (R) of eigenfunctions (ϕ j ) j 1 of A A ϕ j = λ j ϕ j, with λ j 2j 1 and ϕ j h j, where (h j ) j 1 are the Hermite functions. For p 0, we define the Sobolev spaces Let u = X j 1 w j ϕ j H p, then H p = H p (R) = u S (R) : A p/2 u L 2 (R). u 2 H p X j p w j 2. j 1
The result on the nonlinear equation Our main result concerning the nonlinear Schrödinger equation (NLS) is the following Theorem (B. Grébert - LT) Let n 1 be an integer. Then there exist a large class of V S(R) and ε 0 > 0 such that for each ε < ε 0 the solution of (NLS) with initial datum u 0(x) = nx j=1 I 1/2 j e iθ j ϕ j (x), (IC) with (I 1,, I n) (0, 1] n and (θ 1,..., θ n) T n, is quasi-periodic. When θ covers T n, the set of solutions of (NLS) with initial condition (IC) covers a n dimensional torus which is invariant by (NLS). Our result also applies to any non linearity ± u 2m u, with m 1. The set {1,, n} can be replaced by any finite set of N of cardinality n.
The more precise result Let n 1 and Π = [ 1, 1] n. There exist (f k ) 1 k n S(R) such that if we set V (x, ξ) = with ξ = (ξ 1,..., ξ n) Π we have Theorem (B. Grébert - LT) nx ξ k f k (x), j=1 Let n 1 be an integer. Then there exists a Cantor set Π e Π of full measure and ε 0 > 0 so that for each ε < ε 0 and ξ Π e the solution of (NLS) with initial datum u 0(x) = nx j=1 I 1/2 j e iθ j ϕ j (ξ, x), (IC) with (I 1,, I n) (0, 1] n and (θ 1,..., θ n) T n, is quasi-periodic. The solution u reads u(t, x) = nx `Ij + y j (t) 1 2 e iθ j (t) ϕ j (ξ, x) + X z j (t)ϕ j+n (ξ, x). j 1 j=1
Some previous results S.B. Kuksin 93 and J. Pöschel 96 : Case λ j cj d with d > 1 or with smoothing nonlinearity. S.B. Kuksin & J. Pöschel 96 : NLS on [0, π] without external parameter ξ. H. Eliasson & S.B. Kuksin 08 : NLS on T d with V u perturbation. B. Grébert, R. Imekraz & E. Paturel 08 : Normal forms technics. Key ingredient Use of dispersive properties of the Hermite functions (K. Yajima-G. Zhang 01) r > 2, β(r) > 0, h j L r (R) C r j β(r) h j L 2 (R).
The symplectic structure We consider the (complex) Hilbert space l 2 p defined by the norm w 2 p = X j 1 w j 2 j p. We define the symplectic phase space P p as P p = T n R n l 2 p l 2 p (θ, y, z, z), equipped with the canonic symplectic structure nx dθ j dy j + i X dz j dz j. j 1 j=1
The Hamiltonian formulation Let p 2 and n 1. Fix (I 1,..., I n) ]0, 1] n and write 8 < u(x) = P n j=1 (y j + I j ) 1 2 e iθ j ϕ j (ξ, x) + P j 1 z jϕ j+n (ξ, x), : ū(x) = P n j=1 (y j + I j ) 1 2 e iθ j ϕ j (ξ, x) + P j 1 z jϕ j+n (ξ, x), where (θ, y, z, z) P p = T n R n l 2 p l 2 p are regarded as variables. In this setting equation (NLS) reads as the Hamilton equations associated to the Hamiltonian function H = N + P where Λ j (ξ) = λ j+n (ξ) and P(θ, y, z, z) = ε 2 Z R N = nx nx λ j (ξ)y j + X Λ j (ξ)z j z j, j 1 j=1 (y j + I j ) 12 e iθj ϕ j (ξ, x) + X j=1 j 1 nx j=1 (y j + I j ) 12 e iθj ϕ j (ξ, x) + X j 1 z j ϕ j+n (ξ, x), z j ϕ j+n (ξ, x) 4dx.
The Hamiltonian formulation In other words, we obtain the following system, which is equivalent to (NLS) 8 >< >: θ j = H y j, ż j = i H z j, ẏ j = H θ j, 1 j n ż j = i H z j, j 1 (θ j (0), y j (0), z j (0), z j (0)) = (θ 0 j, y 0 j, z 0 j, z 0 j ), where the initial conditions are chosen so that nx u 0(x) = (I j + yj 0 ) 1 2 e iθj 0 ϕ j (ε, x) + X zj 0 ϕ j+n (ξ, x). j=1 j 1
The general KAM strategy Consider a smooth function F = F (θ, y, z, z), and denote by XF t the flow of the equation 8 θ j = F, ẏ y j = F, 1 j n j θ j >< ż j = i F, ż z j = i F, j 1 j z j >: (θ j (0), y j (0), z j (0), z j (0)) = (θ 0 j, y 0 j, z 0 j, z 0 j ). If F is small enough, X 1 F is well defined and we have (i) The application X 1 F preserves the symplectic structure. (ii) For any smooth G we have Idea of the KAM iteration d `G X t F = G, F X t F. dt Find F so that H X 1 F is in a better form than H = N + P.
The general KAM strategy Write the expansion P = X X P kmqq e ik θ y m z q z q, m,q,q k Z n We then consider the second order Taylor approximation of P which is X X R = R kmqq e ik θ y m z q z q, k Z n 2 m + q+q 2 where R kmqq = P kmqq Thanks to the Taylor formula we can write H X 1 F = N X 1 F + R X 1 F + (P R) X 1 F = N + N, F + +R + Z 1 0 Z 1 0 (1 t) N, F, F X t F dt + R, F X t F dt + (P R) X 1 F. Assume that we can find F and b N which has the same form as N and which satisfy the so-called homological equation N, F + R = b N.
The general KAM strategy Once the homological equation is solved : We define the new normal form by N + = N + b N, the frequencies of which are given by where λ + (ξ) = λ(ξ) + b λ(ξ) and Λ + (ξ) = Λ(ξ) + b Λ(ξ), bλ j (ξ) = N b (0, 0, 0, 0, ξ) and Λ y b j (ξ) = 2 N b (0, 0, 0, 0, ξ). j z j z j We define the new perturbation term P + by P + = (P R) X 1 F + Z 1 where R(t) = (1 t) b N + tr in such a way that H X 1 F = N + + P +. 0 R(t), F X t F dt, Convergence : If P = O(ε) and F = O(ε). Then R = O(ε) and the quadratic part of P + is O(ε 2 ).
The general KAM strategy At the end we obtain a symplectic transformation Φ (near the origin) so that H = H Φ = N + P, where N = nx λ j (ξ)y j + X Λ j (ξ)z j z j, j 1 j=1 and P has no quadratic part in z, z and no linear part in y. Then the new coordinates (y, θ, z, z ) = Φ 1 (y, θ, z, z) satisfy 8 >< >: θ j = H, ẏ y j j = H, 1 j n θ j ż j = i H z j, ż j = i H, j 1. z j In particular, the solution to (NH) with initial condition (θ j(0), y j (0), z j (0), z j(0)) = (θ 0, 0, 0, 0) reads j (θ j(t), y j (t), z j (t), z j(t)) = (tλ j + θ 0 j, 0, 0, 0). Hence we have constructed a quasi-periodic solution to (NLS). (NH)
The homological equation Aim : Solve N, F + R = b N, with N = nx λ j (ξ)y j + X Λ j (ξ)z j z j. j 1 j=1 We look for a solution F of the form X X F = F kmqq e ik θ y m z q z q. k Z n A direct computation gives 8 >< if kmqq = >: 2 m + q+q 2 R kmqq, if k + q q 0, k λ(ξ) + (q q) Λ(ξ) 0, otherwise, bn = [R] = X m + q =1 R 0mqqy m z q z q.
Control of the frequencies We show that we can find (f k ) 1 k n such that Small divisors control : There exist a subset Π α Π with Meas(Π\Π α) 0 when α 0 and τ > 1, such that for all ξ Π α k λ(ξ) + l Λ(ξ) α l, (k, l) Z, 1 + k τ where Z := {(k, l) Z n Z, (k, l) 0, l 2}. To perform the KAM method, we now have to check this condition persists after each iteration. This will be the case if the perturbation b Λ j satisfies b Λ j Cεj β for some β > 0.
Control of the frequencies We have bλ j (ξ) = 2 N b (0, 0, 0, 0, ξ) = z j z j In our case, we have P = ε Z u 4, therefore 2 R 2 P z j z j (0, 0, 0, 0, ξ). 2 P z j z j Now by the dispersive estimate ϕ j L (R) Cj 1/12 we get Z = 2ε ϕ 2 j+n u 2. 2 P ε ϕj+n 2 L z j z (R) u 2 L 2 (R) Cεj 1/6 u 2 L 2 (R). j R
Reducibility of the linear equation We consider the linear equation ( i tu + 2 x u x 2 u + ɛv (tω, x)u = 0, (t, x) R R, u(0, x) = u 0(x), (LS) where the potential V : T n R (θ, x) R satisfies V is analytic in θ. V is C in x, with bounded derivatives. V satisfies V (θ, x) C(1 + x 2 ) δ for some δ > 0.
Reducibility of the linear equation ( i tu + 2 x u x 2 u + ɛv (tω, x)u = 0, (t, x) R R, u(0, x) = u 0(x). (LS) Theorem (B. Grébert - LT) There exists ɛ 0 such that for all 0 ɛ < ɛ 0 there exists Λ ε [0, 2π) n such that [0, 2π) n \Λ ε 0 as ɛ 0, and such that for all ω Λ ε, the linear Schrödinger equation (LS) reduces, in L 2 (R), to a linear equation with constant coefficients.
Reducibility of the linear equation ( i tu + 2 x u x 2 u + ɛv (tω, x)u = 0, (t, x) R R, u(0, x) = u 0(x). (LS) Theorem (B. Grébert - LT) There exists ɛ 0 such that for all 0 ɛ < ɛ 0 there exists Λ ε [0, 2π) n such that [0, 2π) n \Λ ε 0 as ɛ 0, and such that for all ω Λ ε, the linear Schrödinger equation (LS) reduces, in L 2 (R), to a linear equation with constant coefficients. Corollary Let ω Λ ε, then any solution u of (LS) is almost-periodic in time and we have the bounds (1 εc) u 0 H p u(t) H p (1 + εc) u 0 H p, t R, for some C = C(p, ω).
Reducibility of the linear equation Some previous results D. Bambusi & S. Graffi 01 ; J. Lui & X. Yuan 10 : Case x β, β > 2. W.-M. Wang 08 : Case x 2, for some particular V. H. Eliasson & S.B. Kuksin 08 : NLS on T d. W.-M. Wang 08, J.-M. Delort 10, D. Fang & Q. Zhang 10 : NLS on T d : Bounds u(t) H p ` ln t cp if V is analytic. J.-M. Delort 10 : Existence of solutions so that u(t) H p t p/2 if V is allowed to be a pseudo-differential operator.
Hamiltonian formulation Equation (LS) reads as a non autonomous Hamiltonian system 8 < ż j = i(2j 1)z j iε z Q(t, e z, z), j 1 j where : z j = i(2j 1) z j + iε z j e Q(t, z, z), j 1 Z X eq(t, z, z) = V (ωt, x)` z j h (x) ` X j z j h j (x) dx. R We re-interpret this system as an autonomous Hamiltonian system in an extended phase space 8 ż j = i(2j 1)z j iε Q(θ, z, z) j 1 z j >< z j = i(2j 1) z j + iε Q(θ, z, z) j 1 z j θ j = ω j j = 1,, n >: ẏ j = ε Q(θ, z, z) j = 1,, n θ j j 1 j 1 where Z X Q(θ, z, z) = V (θ, x)` z j h (x) ` X j z j h j (x) dx. R j 1 j 1
Linear dynamics Here the external parameters are directly the frequencies ω = (ω j ) 1 j n [0, 2π) n =: Π and the normal frequencies Ω j = 2j 1 are constant. Using the KAM scheme, we are able to show the existence of a set of parameters Π ε Π with Π\Π ε 0 when ε 0 and a coordinate transformation Φ : Π ε P 0 P 0, such that H Φ = N, where N takes the form nx N (ω) = Ω j z j z j. j=1 ω j y j + X j 1 In the new coordinates, (y, θ, z, z ) = Φ 1 (y, θ, z, z), the dynamic is linear with y invariant : 8 ż j >< = iω j z j j 1 z j = iω j z j j 1 θ j >: = ω j j = 1,, n ẏ j = 0 j = 1,, n.