KAM for quasi-linear KdV Massimiliano Berti Toronto, 10-1-2014, - PowerPoint PPT Presentation

The problem Literature Main results Proof: forced case Proof: Autonomous case KAM for quasi-linear KdV Massimiliano Berti Toronto, 10-1-2014, Conference on ”Hamiltonian PDEs: Analysis, Computations and Applications” for the 60 -th birthday of Walter Craig

The problem Literature Main results Proof: forced case Proof: Autonomous case 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 { ( ∂ u x 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 , ε u , ε u x , ε u xx , ε u 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 (that I learned from Walter Craig)

The problem Literature Main results Proof: forced case Proof: Autonomous case Hamiltonian PDE u t = X H ( u ) , X H ( u ) := ∂ x ∇ L 2 H ( u ) Hamiltonian KdV � u 2 2 + u 3 + f ( x , u , u x ) dx x H = T 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: � T ( ∂ − 1 Ω( u , v ) := x u ) v dx

The problem Literature Main results Proof: forced case Proof: Autonomous case Goal: look for small amplitude quasi-periodic solutions Definition: quasi-periodic solution with n frequencies u ( t , x ) = U ( ω t , 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 { ω t } 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

The problem Literature Main results Proof: forced case Proof: Autonomous case Linear Airy eq. x ∈ T u t + u xxx = 0 , Solutions: (superposition principle) � a j e i j 3 t e i jx u ( t , x ) = j ∈ Z \{ 0 } Eigenvalues j 3 = " normal frequencies " Eigenfunctions: e i jx = " normal modes " All solutions are 2 π - periodic in time: completely resonant ⇒ Quasi-periodic solutions are a completely nonlinear phenomenon

The problem Literature Main results Proof: forced case Proof: Autonomous case Linear Airy eq. x ∈ T u t + u xxx = 0 , Solutions: (superposition principle) � a j e i j 3 t e i jx u ( t , x ) = j ∈ Z \{ 0 } Eigenvalues j 3 = " normal frequencies " Eigenfunctions: e i jx = " normal modes " All solutions are 2 π - periodic in time: completely resonant ⇒ Quasi-periodic solutions are a completely nonlinear phenomenon

The problem Literature Main results Proof: forced case Proof: Autonomous case KdV is completely integrable u t + u xxx − 3 ∂ x u 2 = 0 All solutions are periodic, quasi-periodic, almost periodic What happens under a small perturbation?

The problem Literature Main results Proof: forced case Proof: Autonomous case 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

The problem Literature Main results Proof: forced case Proof: Autonomous case 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

The problem Literature Main results Proof: forced case Proof: Autonomous case Literature: KAM for "unbounded" perturbations Liu-Yuan ’10 for Hamiltonian DNLS (and Benjamin-Ono) i u t − u xx + M σ u + i ε f ( u , ¯ u ) u x = 0 Zhang-Gao-Yuan ’11 Reversible DNLS i u t + u xx = | u x | 2 u Craig-Wayne periodic solutions, Lyapunov-Schmidt + Nash-Moser Bourgain ’96, Derivative NLW y tt − y xx + my + y 2 t = 0 , m � = 0, Craig ’00, Hamiltonian DNLW � y tt − y xx + g ( x ) y = f ( x , D β y ) , D := − ∂ xx + g ( x ) ,

The problem Literature Main results Proof: forced case Proof: Autonomous case quasi-periodic solutions Berti-Biasco-Procesi ’12, ’13, 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, Water waves: quasi-linear equation, new ideas for conjugation of linearized operator

The problem Literature Main results Proof: forced case Proof: Autonomous case quasi-periodic solutions Berti-Biasco-Procesi ’12, ’13, 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, Water waves: quasi-linear equation, new ideas for conjugation of linearized operator

The problem Literature Main results Proof: forced case Proof: Autonomous case 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 ̺ u := u ( − x ) , ̺ 2 = I , − ̺ X H ( u ) = X H ( ̺ u )

The problem Literature Main results Proof: forced case Proof: Autonomous case 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"  n } ⊂ Z \ { 0 } , S := {− ¯  n , . . . , − ¯  1 , ¯  1 , . . . , ¯ 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 � � � ( ξ ) t e i jx + o ( j ( ξ ) = j 3 + O ( | ξ | ) ξ j e i ω ∞ ξ ) , ω ∞ u = j j ∈ S 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

The problem Literature Main results Proof: forced case Proof: Autonomous case Tangential sites Explicit conditions: Hypothesis ( S 3 ) j 1 + j 2 + j 3 � = 0 for all j 1 , j 2 , j 3 ∈ S Hypothesis ( S 4 ) ∄ j 1 , . . . , j 4 ∈ S such that 4 − ( j 1 + j 2 + j 3 + j 4 ) 3 = 0 j 3 1 + j 3 2 + j 3 3 + j 3 j 1 + j 2 + j 3 + j 4 � = 0 , 1 ( S 3 ) used in the linearized operator. If f 5 = 0 then not needed 2 If also f 6 = 0 then ( S 4 ) not needed (used in Birkhoff-normal-form) “genericity”: After fixing { ¯  n } , in the choice of ¯  n + 1 ∈ N there are only  1 , . . . , ¯ finitely many forbidden values

The problem Literature Main results Proof: forced case Proof: Autonomous case Comments 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 , x ∈ T for all the tangential sites S := {− ¯  n , . . . , − ¯  n }  1 , ¯  1 , . . . , ¯ such that n � 2  2 ∈ N ¯ i / 2 n − 1 i = 1 2 If f = f ( u , u x ) the result is true for all the tangential sites S 3 Also for generalized KdV (not integrable), with normal form techniques of Procesi-Procesi

The problem Literature Main results Proof: forced case Proof: Autonomous case Linear stability (L): linearized equation ∂ t h = ∂ x ∂ u ∇ H ( u ( ω t , x )) h h t + a 3 ( ω t , x ) h xxx + a 2 ( ω t , x ) h xx + a 1 ( ω t , x ) h x + a 0 ( ω t , x ) h = 0 There exists a quasi-periodic (Floquet) change of variable ψ ∈ T ν , η ∈ R ν , v ∈ H s x ∩ L 2 h = Φ( ω t )( ψ, η, v ) , S ⊥ which transforms (L) into the constant coefficients system  ˙  ψ = b η   η = 0 ˙    ∈ S , µ j ∈ R v j = i µ j v j , ˙ j / ⇒ η ( t ) = η 0 , v j ( t ) = v j ( 0 ) e i µ j t = = ⇒ � v ( t ) � s = � v ( 0 ) � s : stability

The problem Literature Main results Proof: forced case Proof: Autonomous case Forced quasi-linear perturbations of Airy ω ∈ R n as 1-dim. parameter Use ω = λ� Theorem (Baldi, Berti, Montalto , to appear Math. Annalen) There exist s := s ( n ) > 0 , q := q ( n ) ∈ N , such that: ω ∈ R n diophantine. For every quasi-linear Hamiltonian Let � 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 + ε f ( λ� ω t , 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 .