KAM for quasi-linear KdV

Transcription

1 KAM for quasi-linear KdV Massimiliano Berti ST Etienne de Tinée,

2 KdV t u + u xxx 3 x u 2 + N 4 (x, u, u x, u xx, u xxx ) = 0, x T Quasi-linear Hamiltonian perturbation N 4 := x {( u f )(x, u, u x )} + xx {( ux f )(x, u, u x )} N 4 = a 0 (x, u, u x, u xx ) + a 1 (x, u, u x, u xx )u xxx N 4 (x, εu, εu x, εu xx, εu xxx ) = O(ε 4 ), ε 0 f (x, u, u x ) = O( u 5 + u x 5 ), f C q (T R R, R) Physically important for perturbative derivation from water-waves Control the effect of N 4 = O(ε 4 xxx ) over infinite times...

3 KdV t u + u xxx 3 x u 2 + N 4 (x, u, u x, u xx, u xxx ) = 0, x T Quasi-linear Hamiltonian perturbation N 4 := x {( u f )(x, u, u x )} + xx {( ux f )(x, u, u x )} N 4 = a 0 (x, u, u x, u xx ) + a 1 (x, u, u x, u xx )u xxx N 4 (x, εu, εu x, εu xx, εu xxx ) = O(ε 4 ), ε 0 f (x, u, u x ) = O( u 5 + u x 5 ), f C q (T R R, R) Physically important for perturbative derivation from water-waves Control the effect of N 4 = O(ε 4 xxx ) over infinite times...

4 Hamiltonian PDE u t = X H (u), Hamiltonian KdV H = T X H (u) := x L 2H(u) u 2 x 2 + u3 + f (x, u, u x )dx where the density f (x, u, u x ) = O( (u, u x ) 5 ) Phase space H 1 0 (T) := {u(x) H 1 (T, R) : T u(x)dx = 0 } Non-degenerate symplectic form: Ω(u, v) := T ( 1 x u) v dx

5 Goal: look for small amplitude quasi-periodic solutions Definition: quasi-periodic solution with n frequencies u(t, x) = U(ωt, x) where U(ϕ, x) : T n T R, ω R n (= frequency vector) is irrational ω k 0, k Z n \ {0} = the linear flow {ωt} t R is dense on T n The torus-manifold T n ϕ u(ϕ, x) phase space is invariant under the flow evolution of the PDE

6 Linear Airy eq. u t + u xxx = 0, x T Solutions: (superposition principle) u(t, x) = j Z\{0} a j e ij3t e ijx Eigenvalues j 3 = "normal frequencies" Eigenfunctions: e ijx = "normal modes" All solutions are 2π- periodic in time: completely resonant Quasi-periodic solutions are a completely nonlinear phenomenon

7 Linear Airy eq. u t + u xxx = 0, x T Solutions: (superposition principle) u(t, x) = j Z\{0} a j e ij3t e ijx Eigenvalues j 3 = "normal frequencies" Eigenfunctions: e ijx = "normal modes" All solutions are 2π- periodic in time: completely resonant Quasi-periodic solutions are a completely nonlinear phenomenon

8 KdV is completely integrable u t + u xxx 3 x u 2 = 0 Moreover Kappeler proved that it has Birkhoff-coordinates All solutions are periodic, quasi-periodic, almost periodic QUESTION: What happens adding a small perturbation?

9 KdV is completely integrable u t + u xxx 3 x u 2 = 0 Moreover Kappeler proved that it has Birkhoff-coordinates All solutions are periodic, quasi-periodic, almost periodic QUESTION: What happens adding a small perturbation?

10 KAM theory Kuksin 98, Kappeler-Pöschel 03: KAM for KdV u t + u xxx + uu x + ε x f (x, u) = 0 1 semilinear perturbation x f (x, u) 2 Also true for Hamiltonian perturbations u t + u xxx + uu x + ε x x 1/2 f (x, x 1/2 u) = 0 of order 2 j 3 i 3 i 2 + j 2, i j = KdV gains up to 2 spatial derivatives 3 for quasi-linear KdV? OPEN PROBLEM

11 KAM theory Kuksin 98, Kappeler-Pöschel 03: KAM for KdV u t + u xxx + uu x + ε x f (x, u) = 0 1 semilinear perturbation x f (x, u) 2 Also true for Hamiltonian perturbations u t + u xxx + uu x + ε x x 1/2 f (x, x 1/2 u) = 0 of order 2 j 3 i 3 i 2 + j 2, i j = KdV gains up to 2 spatial derivatives 3 for quasi-linear KdV? OPEN PROBLEM

12 Literature: KAM for "unbounded" perturbations Liu-Yuan 10 for Hamiltonian DNLS (and Benjamin-Ono) iu t u xx + M σ u + iε f (u, ū)u x = 0 Zhang-Gao-Yuan 11 Reversible DNLS iu t + u xx = u x 2 u Less dispersive = more difficult Bourgain 96, Derivative NLW, periodic solutions y tt y xx + my + yt 2 = 0, m 0,

13 Existence and stability of quasi-periodic solutions: Berti-Biasco-Procesi, Ann. Sci. ENS 13, Hamiltonian DNLW y tt y xx + my + g(dy) = 0, D := xx + m, Berti-Biasco-Procesi, Arch.Rat.Mech.Anal 14, Reversible DNLW u tt u xx + mu = g(x, u, u x, u t ) For quasi-linear PDEs: Periodic solutions: Iooss-Plotinikov-Toland, Iooss-Plotnikov, 01-10, Gravity Water Waves with Finite or Infinite depth, New ideas for conjugation of linearized operator Baldi 12, quasi-linear and fully non-linear Benjiamin-Ono

14 Existence and stability of quasi-periodic solutions: Berti-Biasco-Procesi, Ann. Sci. ENS 13, Hamiltonian DNLW y tt y xx + my + g(dy) = 0, D := xx + m, Berti-Biasco-Procesi, Arch.Rat.Mech.Anal 14, Reversible DNLW u tt u xx + mu = g(x, u, u x, u t ) For quasi-linear PDEs: Periodic solutions: Iooss-Plotinikov-Toland, Iooss-Plotnikov, 01-10, Gravity Water Waves with Finite or Infinite depth, New ideas for conjugation of linearized operator Baldi 12, quasi-linear and fully non-linear Benjiamin-Ono

15 Main results Hamiltonian density: f (x, u, u x ) = f 5 (u, u x ) + f 6 (x, u, u x ) f 5 polynomial of order 5 in (u, u x ); f 6 (x, u, u x ) = O( u + u x ) 6 Reversibility condition: f (x, u, u x ) = f ( x, u, u x ) KdV-vector field X H (u) := x H(u) is reversible w.r.t the involution ϱu := u( x), ϱ 2 = I, ϱx H (u) = X H (ϱu)

16 Theorem ( 13, P. Baldi, M. Berti, R. Montalto) Let f C q (with q := q(n) large enough). Then, for generic choice of the "tangential sites" S := { j n,..., j 1, j 1,..., j n } Z \ {0}, the hamiltonian and reversible KdV equation t u + u xxx 3 x u 2 + N 4 (x, u, u x, u xx, u xxx ) = 0, x T, possesses small amplitude quasi-periodic solutions with Sobolev regularity H s, s q, of the form u = ξ j e iω j (ξ) t e ijx + o( ξ), ω j S j (ξ) = j 3 + O( ξ ) for a "Cantor-like" set of "initial conditions" ξ R n with density 1 at ξ = 0. The linearized equations at these quasi-periodic solutions are reduced to constant coefficients and are stable. If f = f 7 = O( (u, u x ) 7 ) then any choice of tangential sites

17 1 A similar result holds for mkdv: focusing/defocusing t u + u xxx ± x u 3 + N 4 (x, u, u x, u xx, u xxx ) = 0 for tangential sites S := { j n,..., j 1, j 1,..., j n } such that 2 2n 1 n i=1 j 2 i / N 2 If f = f (u, u x ) the result is true for all the tangential sites S 3 Also for generalized KdV (not integrable) generalized KdV t u + u xxx ± x u p + N (x, u, u x, u xx, u xxx ) = 0 by normal form techniques of Procesi-Procesi

18 Linear stability (L): linearized equation t h = x u H(u(ωt, x))h h t + a 3 (ωt, x)h xxx + a 2 (ωt, x)h xx + a 1 (ωt, x)h x + a 0 (ωt, x)h = 0 There exists a quasi-periodic (Floquet) change of variable h = Φ(ωt)(ψ, η, v), ψ T ν, η R ν, v H s x L 2 S which transforms (L) into the constant coefficients system ψ = bη η = 0 v j = iµ j v j, j / S, µ j R = η(t) = η 0, v j (t) = v j (0)e iµ jt = v(t) s = v(0) s : stability

19 Forced quasi-linear perturbations of Airy Use ω = λ ω R n as 1-dim. parameter Theorem (Baldi, Berti, Montalto, to appear Math. Annalen) There exist s := s(n) > 0, q := q(n) N, such that: Let ω R n diophantine. For every quasi-linear Hamiltonian nonlinearity f C q for all ε (0, ε 0 ) small enough, there is a Cantor set C ε [1/2, 3/2] of asymptotically full measure, i.e. C ε 1 as ε 0, such that for all λ C ε the perturbed Airy equation t u + xxx u + εf (λ ωt, x, u, u x, u xx, u xxx ) = 0 has a quasi-periodic solution u(ε, λ) H s (for some s q) with frequency ω = λ ω and satisfying u(ε, λ) s 0 as ε 0.

20 Key: spectral analysis of quasi-periodic operator L = ω ϕ + xxx +a 3 (ϕ, x) xxx +a 2 (ϕ, x) xx +a 1 (ϕ, x) x +a 0 (ϕ, x) a i = O(ε), i = 0, 1, 2, 3 Main problem: the non constant coefficients term a 3 (ϕ, x) xxx! Main difficulties: 1 The usual KAM iterative scheme is unbounded 2 We expect an estimate of perturbed eigenvalues µ j (ε) = j 3 + O(εj 3 ) which is NOT sufficient for verifying second order Melnikov ω l + µ j (ε) µ i (ε) γ j3 i 3 l τ, l, j, i

21 Idea to conjugate L to a diagonal operator 1 "REDUCTION IN DECREASING SYMBOLS" L 1 := Φ 1 LΦ = ω ϕ + m 3 xxx + m 1 x + R 0 R 0 (ϕ, x) pseudo-differential operator of order 0, R 0 (ϕ, x) : H s x H s x, R 0 = O(ε), m 3 = 1 + O(ε), m 1 = O(ε), m 1, m 3 R, constants Use suitable transformations far from the identity 2 "REDUCTION OF THE SIZE of R 0 " L ν := Φ 1 ν L 1 Φ ν = ω ϕ + m 3 xxx + m 1 x + r (ν) + R ν 2 R ν = R ν (ϕ, x) = O(R ν 0 ) r (ν) = diag j Z (r (ν) j ), sup j r (ν) = O(ε), KAM-type scheme, now transformations of H s x H s x j

22 Idea to conjugate L to a diagonal operator 1 "REDUCTION IN DECREASING SYMBOLS" L 1 := Φ 1 LΦ = ω ϕ + m 3 xxx + m 1 x + R 0 R 0 (ϕ, x) pseudo-differential operator of order 0, R 0 (ϕ, x) : H s x H s x, R 0 = O(ε), m 3 = 1 + O(ε), m 1 = O(ε), m 1, m 3 R, constants Use suitable transformations far from the identity 2 "REDUCTION OF THE SIZE of R 0 " L ν := Φ 1 ν L 1 Φ ν = ω ϕ + m 3 xxx + m 1 x + r (ν) + R ν 2 R ν = R ν (ϕ, x) = O(R ν 0 ) r (ν) = diag j Z (r (ν) j ), sup j r (ν) = O(ε), KAM-type scheme, now transformations of H s x H s x j

23 Higher order term L := ω ϕ + xxx + εa 3 (ϕ, x) xxx STEP 1: Under the symplectic change of variables (Au) := (1 + β x (ϕ, x))u(ϕ, x + β(ϕ, x)) we get L 1 := A 1 LA = ω ϕ + (A 1 (1 + εa 3 )(1 + β x ) 3 ) xxx + O( xx ) = ω ϕ + c(ϕ) xxx + O( xx ) imposing (1 + εa 3 )(1 + β x ) 3 = c(ϕ), There exist solution c(ϕ) 1, β = O(ε)

24 STEP 2: Rescaling time (Bu)(ϕ, x) = u(ϕ + ωq(ϕ), x) we have B 1 L 1 B = B 1 (1 + ω ϕ q)(ω ϕ ) + B 1 c(ϕ) xxx + O( xx ) = µ(ε)b 1 c(ϕ)(ω ϕ ) + B 1 c(ϕ) xxx + O( xx ) solving 1 + ω ϕ q = µ(ε)c(ϕ), q(ϕ) = O(ε) Dividing for µ(ε)b 1 c(ϕ) we get L 2 := ω ϕ + m 3 (ε) xxx + O( x ), m 3 (ε) := µ 1 (ε) = 1 + O(ε) which has the leading order with constant coefficients

25 Autonomous KdV New further difficulties: 1 No external parameters. The frequency of the solutions is not fixed a-priori. Frequency-amplitude modulation. 2 KdV is completely resonant 3 Construction of an approximate inverse Ideas: 1 Weak Birkhoff-normal form 2 General method to decouple the "tangential dynamics" from the "normal dynamics", developed with P. Bolle Procedure which reduces autonomous case to the forced one

26 Step 1. Bifurcation analysis: weak Birkhoff normal form Fix the tangential sites S := { j n,..., j 1, j 1,..., j n } Z \ {0} Split the dynamics: Hamiltonian H = 1 vx T 2 + v 3 dx + 3 T u(x) = v(x) + z(x) v(x) = j S u je ijx = tangential component z(x) = j / S u je ijx = normal component T T zx 2 dx + v 2 zdx + 3 T v 3 dx + 3 T vz 2 dx + T v 2 zdx z 3 dx + T T f (u, u x ) Goal: eliminate terms linear in z = {z = 0} is invariant manifold

27 Theorem (Weak Birkhoff normal form) There is a symplectic transformation Φ B : H 1 0 (T x) H 1 0 (T x) Φ B (u) = u + Ψ(u), Ψ(u) = Π E Ψ(Π E u), where E := span{e ijx, 0 < j 6 S } is finite-dimensional, s.t. H 3 := T H 4,2 := 6 T H := H Φ B = H 2 + H 3 + H 4 + H 5 + H 6, z 3 dx + 3 vz 2 dx, H 4 := 3 u j 4 T 2 j 2 + H 4,2 + H 4,3 j S ( vzπ S ( 1 x v)( x 1 z) ) dx + 3 H 4,3 := R(vz 3 ), H 5 := 5 T z 2 π 0 ( 1 x v) 2 dx, q=2 R(v 5 q z q ), and H 6 collects all the terms of order at least six in (v, z).

28 Fourier representation u(x) = j Z\{0} u j e ijx, u(x) (u j ) j Z\{0} First Step. Eliminate the u j1 u j2 u j3 of H 3 with at most one index outside S. Since j 1 + j 2 + j 3 = 0 they are finitely many Φ := the time 1-flow map generated by F (u) := F j1,j 2,j 3 u j1 u j2 u j3 j 1 +j 2 +j 3 =0 The vector field X F is supported on finitely many sites X F (u) = Π H2S X F ( ΠH2S u ) = the flow is a finite dimensional perturbation of the identity Φ = Id + Ψ, Ψ = Π H2S ΨΠ H2S

29 For the other steps: Normalize the quartic monomials u j1 u j2 u j3 u j4, j 1, j 2, j 3, j 4 S. The fourth order system H 4 restricted to S turns out to be integrable, i.e. 3 2 u j 4 j S j 2 (non isochronous rotators) Now {z = 0} is an invariant manifold for H 4 filled by quasi-periodic solutions with a frequency which varies with the amplitude

30 Difference w.r.t. other Birkhoff normal forms 1 Kappeler-Pöschel (KdV), Kuksin-Pöschel (NLS), complete Birkhoff-normal form: they remove/normalize also the terms O(z 2 ), O(z 3 ), O(z 4 ) 2 Pöschel (NLW), semi normal Birkhoff normal form: normalized only the term O(z 2 ) 3 Kappeler Global Birkhoff normal form for KdV, 1-d-cubic-NLS The above transformations are (1) I + bounded, (2) I + O( 1 x ), (3) Φ = F + O( 1 x ), It is not enough for quasi-linear perturbations! Our Φ = Id + finite dimensional = it changes very little the third order differential perturbations in KdV

31 Rescaled action-angle variables: u := εv ε (θ, y) + εz := ε j S ξ j + j y j e iθ j e ijx + εz Hamiltonian: H ε = N + P, N (θ, y, z, ξ) = α(ξ) y + 1 ( ) 2 N(θ, ξ)z, z where Frequency-amplitude map: α(ξ) = ω + ε 2 Aξ Variable coefficients normal form: ( ) N(θ, ξ)z, z 1 2 L 2 (T) = 1 2( ( z H ε )(θ, 0, 0)[z], z ) L 2 (T) L 2 (T)

32 We look for quasi-periodic solutions of X Hε with Diophantine frequencies: ω = ω + ε 2 Aξ Embedded torus equation: Functional setting F(ε, X) := ω i(ϕ) X Hε (i(ϕ)) = 0 ω θ(ϕ) y H ε (i(ϕ)) ω y(ϕ) + θ H ε (i(ϕ)) ω z(ϕ) x z H ε (i(ϕ)) = 0 unknown: X := i(ϕ) := (θ(ϕ), y(ϕ), z(ϕ))

33 Main Difficulty Invert linearized operator at approximate solution i 0 (ϕ): D i F(i 0 (ϕ))[î] = ω θ θy H ε (i 0 )[ θ] yy H ε (i 0 )[ŷ] zy H ε (i 0 )[ẑ] ω ŷ + θθ H ε (i 0 )[ θ] + θy H ε (i 0 )[ŷ] + θz H ε (i 0 )[ẑ] ω ẑ x { θ z H ε (i 0 )[ θ] + y z H ε [ŷ] + z z H ε [ẑ] }

34 Approximate inverse. Zehnder A linear operator T (X), X := i(ϕ) is an approximate inverse of df (X) if df (X)T (X) Id F (X) 1 T (X) is an exact inverse of df (X) at a solution 2 It is sufficient to invert df (X) at a solution Use the general method to construct an approximate inverse, reducing to the inversion of quasi-periodically forced systems, Berti-Bolle for autonomous NLS-NLW with multiplicative potential

35 How to take advantage that i 0 is a solution? The invariant torus i 0 (ϕ) := (θ 0 (ϕ), y 0 (ϕ), z 0 (ϕ)) is ISOTROPIC = the transformation G of the phase space T n R n H S θ ψ θ 0 (ψ) y := G η := y 0 (ψ) + Dθ 0 (ψ) T η + D z 0 (θ 0 (ψ)) T 1 x w z w z 0 (ψ) + w where z 0 (θ) := z 0 (θ0 1 (θ)), is SYMPLECTIC In the new symplectic coordinates, i 0 is the trivial embedded torus (ψ, η, w) = (ϕ, 0, 0)

36 Transformed Hamiltonian K := H ε G =K 00 (ψ) + K 10 (ψ)η + (K 01 (ψ), w) L 2 x K 20(ψ)η η + ( K 11 (ψ)η, w ) L 2 x + 1 K02 (ψ)w, w 2( ) + O( η + w ) 3 L 2 x Hamiltonian system in new coordinates: ψ = K 10 (ψ) + K 20 (ψ)η + K11 T (ψ)w + O(η2 + w 2 ) η = ψ K 00 (ψ) ψ K 10 (ψ)η ψ K 01 (ψ)w + O(η 2 + w 2 ) ( ẇ = x K01 (ψ) + K 11 (ψ)η + K 02 (ψ)w ) + O(η 2 + w 2 ) Since (ψ, η, w) = (ωt, 0, 0) is a solution = ψ K 00 (ψ) = 0, K 10 (ψ) = ω, K 01 (ψ) = 0

37 = KAM (variable coefficients) normal-form K := H ε G = const + ω η K 20(ψ)η η + ( K 11 (ψ)η, w ) L 2 x + 2( 1 K02 (ψ)w, w ) + O( η + w ) 3 L 2 x Hamiltonian system in new coordinates: ψ = ω + K 20 (ψ)η + K11 T (ψ)w + O(η2 + w 2 ) η = O(η 2 + w 2 ) ( ẇ = x K11 (ψ)η + K 02 (ψ)w ) + O(η 2 + w 2 ) = in the NEW variables the linearized equations at (ϕ, 0, 0) simplify!

38 Linearized equations at the invariant torus (ϕ, 0, 0) ψ ω K 20 (ϕ) η K11 T (ϕ)ŵ a ω η = b ω ŵ x K 11 (ϕ) η x K 02 (ϕ)ŵ c may be solved in a TRIANGULAR way Step 1: solve second equation η = 1 ω b + η 0, η 0 R ν Remark: b has zero average by reversibility, η 0 fixed later Step 2: solve third equation L ω ŵ = c + x K 11 (ϕ) η, L ω := ω ϕ x K 02 (ϕ), This is a quasi-periodically forced linear KdV operator!

39 Reduction of the linearized op. on the normal directions L ω h = Π S ( ( ω ϕ h + xx (a 1 x h)+ x a0 h ) ) ε 2 x R 2 [h] x R [h] a 1 1 := O(ε 3 ), a 0 := εp 1 + ε 2 p The remainders R 2, R are finite range (very regularizing!) Reduce L ω to constant coefficients as in forced case, hence invert it 1 Terms O(ε), O(ε 2 ) are not perturbative: εγ 1, ε 2 γ 1 is large! γ = o(ε 2 ) 2 These terms eliminated by algebraic arguments (integrability property of Birkhoff normal form)

40 Step 3: solve first equation Since ψ ω = K 20 (ϕ) η + K11 T (ϕ)ŵ a K 20 (ϕ) = 3ε 2 Id + o(ε 2 ) the matrix K 20 is invertible and we choose η 0 (the average of η) so that the right hand side has zero average. Hence ) ψ = ω (K 1 20 (ϕ) η + K11 T (ϕ)ŵ a This completes the construction of an approximate inverse