KdV equation with almost periodic initial data Milivoje Lukic (Rice - PowerPoint PPT Presentation

KdV equation with almost periodic initial data Milivoje Lukic (Rice University) joint work with Ilia Binder, David Damanik, Michael Goldstein QMATH13 October 8, 2016

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity KdV equation with almost periodic initial data Consider the initial value problem for the KdV equation: ∂ t u − 6 u ∂ x u + ∂ 3 x u = 0 u ( x , 0) = V ( x ) Theorem (McKean–Trubowitz 1976) If V ∈ H n ( T ) , then there is a global solution u ( x , t ) on T × R and this solution is H n ( T ) -almost periodic in t. This means that u ( · , t ) = F ( ζ t ) for some continuous F : T ∞ → H n ( T ) and ζ ∈ R ∞ . Solutions on T are periodic solutions on R , which motivates the following: Conjecture (Deift 2008) If V : R → R is almost periodic, then there is a global solution u ( x , t ) that is almost periodic in t. Even short time existence of solutions is not known in this generality.

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity Global existence, uniqueness, and almost periodicity The following theorem solves Deift’s conjecture under certain assumptions: Theorem (Binder–Damanik–Goldstein–Lukic) If V : R → R is almost periodic, H V = − ∂ 2 x + V has σ ac ( H V ) = σ ( H V ) = S, and S is “thick enough”, then (existence) there exists a global solution u ( x , t ) ; 1 (uniqueness) if ˜ u is another solution on R × [ − T , T ] , and 2 u , ∂ 3 u ∈ L ∞ ( R × [ − T , T ]) , ˜ x ˜ then ˜ u = u; (x-dependence) for each t, x �→ u ( x , t ) is almost periodic in x; 3 (t-dependence) t �→ u ( · , t ) is W 4 , ∞ ( R ) -almost periodic in t. 4 Thickness conditions will be described below.

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity Application to quasi-periodic initial data An explicit class of almost periodic initial data covered by this result is the following. Consider a quasi-periodic potential given by V ( x ) = U ( ω x ) with sampling function U : T ν → R and frequency vector ω ∈ R ν . Assume that the sampling function is small and analytic: � c ( m ) e 2 π im θ U ( θ ) = m ∈ Z ν | c ( m ) | ≤ ε e − κ 0 | m | for some ε > 0, 0 < κ 0 ≤ 1. We also assume that the frequency vector ω ∈ R ν is Diophantine, m ∈ Z ν \ { 0 } | m ω | ≥ a 0 | m | − b 0 , for some 0 < a 0 < 1 , ν < b 0 < ∞ . Then the above theorem applies as long as ε < ε 0 ( a 0 , b 0 , κ 0 ).

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity Application to quasi-periodic initial data Theorem If V is quasi-periodic with a Diophantine frequency vector and a sufficiently small analytic sampling function, then (existence) there exists a global solution u ( x , t ) ; 1 (uniqueness) if ˜ u is another solution on R × [ − T , T ] , and 2 u ∈ L ∞ ( R × [ − T , T ]) , u , ∂ 3 ˜ x ˜ then ˜ u = u; (x-dependence) for each t, u ( · , t ) is quasi-periodic in x, 3 � c ( m , t ) e 2 π im θ u ( x , t ) = m ∈ Z ν √ 4 ε e − κ 0 4 | m | | c ( m , t ) | ≤ (t-dependence) t �→ u ( · , t ) is W k , ∞ ( R ) -almost periodic in t, for any 4 integer k ≥ 0 .

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity Reflectionless operators and Remling’s theorem Define Green’s function of H W = − ∂ 2 x + W by G ( x , y ; z ) = � δ x , ( H W − z ) − 1 δ y � W is reflectionless if Re G (0 , 0; E + i 0) = 0 for Lebesgue-a.e. E ∈ S = σ ( H W ) Write W ∈ R ( S ) in this case Theorem (Remling 2007) Assume W is almost periodic and S = σ ( H W ) = σ ac ( H W ) . Then W ∈ R ( S ) . Theorem (Rybkin 2008) Assume that V ∈ R ( S ) and σ ac ( H V ) = S. Assume that u ( x , t ) is a solution such that u , ∂ 3 x u ∈ L ∞ ( R × [ − T , T ]) for some T > 0 . Then, u ( · , t ) ∈ R ( S ) for every t ∈ [ − T , T ] .

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity Torus of Dirichlet data � ( E − j , E + Write the spectrum as S = [ E , ∞ ) \ j ) j ∈ J Fix a gap ( E − j , E + j ) and x ∈ R  G ( x , x ; E ) = 0 , where E ∈ ( E − j , E + E j )   E − G ( x , x ; E ) > 0 , ∀ E ∈ ( E − Define µ j ( x ) = j , E + j ) j G ( x , x ; E ) < 0 , ∀ E ∈ ( E −  E + j , E + j )  j If µ j ( x ) ∈ ( E − j , E + j ), define σ j ( x ) ∈ {±} , so that µ j ( x ) is a Dirichlet eigenvalue of H on [ x , σ j ( x ) ∞ ) � View ( µ j ( x ) , σ j ( x )) j ∈ J as an element of a torus D ( S ) = T j j ∈ J Introduce angular variables ϕ j ( x ) ∈ R / 2 π Z by µ j = E − j − E − + ( E + j ) cos 2 ( ϕ j / 2) j σ j = sgn sin ϕ j

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity The Dubrovin flow and the trace formula Theorem (Craig 1989) Under suitable conditions on S, the ϕ j ( x ) evolve according to the Dubrovin flow d dx ϕ ( x ) = Ψ( ϕ ( x )) which is given by a Lipshitz vector field Ψ , � ( E − k − µ j )( E + � k − µ j ) j − µ j )( E − � � � 4( E − µ j )( E + Ψ j ( ϕ ) = σ j − µ j ) , j ( µ k − µ j ) 2 k � = j and the trace formula recovers the potential, � ( E + j + E − V ( x ) = Q 1 ( ϕ ( x )) := E + − 2 µ j ( x )) . j j ∈ J

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity KdV evolution on Dirichlet data Add time dependence: consider a solution u ( x , t ) and its Dirichlet data µ ( x , t ). Proposition Under suitable “Craig-type” conditions on S, ∂ x ϕ ( x , t ) = Ψ( ϕ ( x , t )) , ∂ t ϕ ( x , t ) = Ξ( ϕ ( x , t )) , where Ξ is a Lipshitz vector field given by Ξ j = − 2( Q 1 + 2 µ j )Ψ j , and the trace formula recovers the solution, � ( E + j + E − u ( x , t ) = Q 1 ( ϕ ( x , t )) = E + − 2 µ j ( x , t )) . j j ∈ J

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity Existence of solutions Under the Craig-type conditions on S , we prove Proposition Let f ∈ D ( S ) . There exists ϕ : R 2 → D ( S ) such that ϕ (0 , 0) = f and ∂ x ϕ ( x , t ) = Ψ( ϕ ( x , t )) , ∂ t ϕ ( x , t ) = Ξ( ϕ ( x , t )) . If we define u : R 2 → R by u ( x , t ) = Q 1 ( ϕ ( x , t )) then the function u ( x , t ) obeys the KdV equation. Moreover, for each t ∈ R , we have u ( · , t ) ∈ R ( S ) and B ( u ( · , t )) = ϕ (0 , t ) . Moreover, if we define Q k = E k + � j ) k + ( E + j ) k − 2 µ k j ∈ J (( E − j ) , then Q 2 ◦ ϕ = − 1 2 ∂ 2 x u + u 2 Q 3 ◦ ϕ = 3 x u − 3 x u − 15 16( ∂ x u ) 2 + u 3 16 ∂ 4 2 u ∂ 2 Proof is by showing convergence of approximants with finite gap spectra S N = [ E , ∞ ) \ � N j =1 ( E − j , E + j ), for which the above statements were known.

KdV equation Reflectionless operators and uniqueness Existence and almost periodicity Almost periodicity of the solution Define ξ j ( z ) as the solution of the Dirichlet problem on C \ S with boundary values on ¯ S given by � x = ∞ or x ∈ S , x ≥ E + 1 j ξ j ( x ) = x ∈ S , x ≤ E − 0 j Sodin–Yuditskii define the infinite dimensional Abel map A : D ( S ) → T J , � σ k ( ξ j ( µ k ) − ξ j ( E − A j ( ϕ ) = π k )) (mod 2 π Z ) k ∈ J Proposition The map A linearizes the KdV flow: for some δ, ζ ∈ R J , A ( ϕ ( x , t )) = A ( ϕ (0 , 0)) + δ x + ζ t . The proof uses finite gap approximants, for which linearity is known, A N j ( ϕ N ( x , t )) = A N j ( ϕ N (0 , 0)) + δ N j x + ζ N j t , and uniform convergence on compacts.

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