KdV equation with almost periodic initial data

Transcription

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

2 KdV equation with almost periodic initial data Consider the initial value problem for the KdV equation: tu 6u xu + x 3 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 (ζt) for some continuous F : T H n (T) and ζ R. Solutions on T are periodic solutions on R, which motivates the following: Conecture (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.

3 Global existence, uniqueness, and almost periodicity The following theorem solves Deift s conecture 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 1 (existence) there exists a global solution u(x, t); 2 (uniqueness) if ũ is another solution on R [ T, T ], and then ũ = u; ũ, 3 x ũ L (R [ T, T ]), 3 (x-dependence) for each t, x u(x, t) is almost periodic in x; 4 (t-dependence) t u(, t) is W 4, (R)-almost periodic in t. Thickness conditions will be described below.

4 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(ωx) with sampling function U : T ν R and frequency vector ω R ν. Assume that the sampling function is small and analytic: U(θ) = m Z ν c(m)e 2πimθ for some ε > 0, 0 < κ 0 1. c(m) εe κ 0 m We also assume that the frequency vector ω R ν is Diophantine, for some 0 < a 0 < 1, ν < b 0 <. mω a 0 m b 0, m Z ν \ {0} Then the above theorem applies as long as ε < ε 0(a 0, b 0, κ 0).

5 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 1 (existence) there exists a global solution u(x, t); 2 (uniqueness) if ũ is another solution on R [ T, T ], and then ũ = u; ũ, 3 x ũ L (R [ T, T ]), 3 (x-dependence) for each t, u(, t) is quasi-periodic in x, u(x, t) = m Z ν c(m, t)e 2πimθ c(m, t) 4ε e κ 0 4 m 4 (t-dependence) t u(, t) is W k, (R)-almost periodic in t, for any integer k 0.

6 Reflectionless operators and Remling s theorem Define Green s function of H W = x 2 + W by G(x, y; z) = δ x, (H W z) 1 δ y W is reflectionless if Re G(0, 0; E + i0) = 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 ].

7 Torus of Dirichlet data Write the spectrum as S = [E, ) \ (E, E + J Fix a gap (E, E + ) and x R E Define µ (x) = E E + ) G(x, x; E) = 0, where E (E, E + G(x, x; E) > 0, E (E, E + ) G(x, x; E) < 0, E (E, E + ) If µ (x) (E, E + ), define σ (x) {±}, so that µ (x) is a Dirichlet eigenvalue of H on [x, σ (x) ) View (µ (x), σ (x)) J as an element of a torus D(S) = J T Introduce angular variables ϕ (x) R/2πZ by µ = E + (E + E ) cos 2 (ϕ /2) σ = sgn sin ϕ )

8 The Dubrovin flow and the trace formula Theorem (Craig 1989) Under suitable conditions on S, the ϕ (x) evolve according to the Dubrovin flow d ϕ(x) = Ψ(ϕ(x)) dx which is given by a Lipshitz vector field Ψ, Ψ (ϕ) = σ 4(E µ )(E + µ )(E and the trace formula recovers the potential, V (x) = Q 1(ϕ(x)) := E + J µ ) k (E + + E (E k µ )(E + k µ ) (µ k µ ) 2, 2µ (x)).

9 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, xϕ(x, t) = Ψ(ϕ(x, t)), tϕ(x, t) = Ξ(ϕ(x, t)), where Ξ is a Lipshitz vector field given by Ξ = 2(Q 1 + 2µ )Ψ, and the trace formula recovers the solution, u(x, t) = Q 1(ϕ(x, t)) = E + J (E + + E 2µ (x, t)).

10 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 xϕ(x, t) = Ψ(ϕ(x, t)), tϕ(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 ((E ) k + (E + ) k 2µ k ), then Q 2 ϕ = x u + u 2 Q 3 ϕ = x u 3 2 u 2 x u ( xu)2 + u 3 Proof is by showing convergence of approximants with finite gap spectra S N = [E, ) \ N =1 (E, E + ), for which the above statements were known.

11 Almost periodicity of the solution Define ξ (z) as the solution of the Dirichlet problem on C \ S with boundary values on S given by { 1 x = or x S, x E + ξ (x) = 0 x S, x E Sodin Yuditskii define the infinite dimensional Abel map A : D(S) T J, A (ϕ) = π k J σ k (ξ (µ k ) ξ (E k )) (mod 2πZ) Proposition The map A linearizes the KdV flow: for some δ, ζ R J, A(ϕ(x, t)) = A(ϕ(0, 0)) + δx + ζt. The proof uses finite gap approximants, for which linearity is known, A N (ϕ N (x, t)) = A N (ϕ N (0, 0)) + δ N x + ζ N t, and uniform convergence on compacts.

12 Thank you!