Symplectic non-squeezing for the discrete nonlinear Schr¨ odinger equation Alexander Tumanov University of Illinois at Urbana-Champaign “Quasilinear equations, inverse problems and their applications” dedicated to the memory of Gennadi Henkin Dolgoprudny, Russia, September 12–15, 2016. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Joint work with Alexandre Sukhov Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Gromov’s Non-Squeezing Theorem Let B n be the unit ball in C n ; then D = B 1 ⊂ C is the unit disc. B n ( r ) is the ball of radius r . Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Gromov’s Non-Squeezing Theorem Let B n be the unit ball in C n ; then D = B 1 ⊂ C is the unit disc. B n ( r ) is the ball of radius r . Let ω = � n � n j = 1 dx j ∧ dy j = i j = 1 dz j ∧ dz j be the standard 2 symplectic form in C n = R 2 n . Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Gromov’s Non-Squeezing Theorem Let B n be the unit ball in C n ; then D = B 1 ⊂ C is the unit disc. B n ( r ) is the ball of radius r . Let ω = � n � n j = 1 dx j ∧ dy j = i j = 1 dz j ∧ dz j be the standard 2 symplectic form in C n = R 2 n . A smooth map F : Ω ⊂ C n → C n is called symplectic if it preserves the symplectic form ω , that is, F ∗ ω = ω . Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Gromov’s Non-Squeezing Theorem Let B n be the unit ball in C n ; then D = B 1 ⊂ C is the unit disc. B n ( r ) is the ball of radius r . Let ω = � n � n j = 1 dx j ∧ dy j = i j = 1 dz j ∧ dz j be the standard 2 symplectic form in C n = R 2 n . A smooth map F : Ω ⊂ C n → C n is called symplectic if it preserves the symplectic form ω , that is, F ∗ ω = ω . Theorem (Gromov, 1985) Let r , R > 0 . Suppose there is a symplectic embedding F : B n ( r ) → D ( R ) × C n − 1 . Then r ≤ R. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Gromov’s Non-Squeezing Theorem Let B n be the unit ball in C n ; then D = B 1 ⊂ C is the unit disc. B n ( r ) is the ball of radius r . Let ω = � n � n j = 1 dx j ∧ dy j = i j = 1 dz j ∧ dz j be the standard 2 symplectic form in C n = R 2 n . A smooth map F : Ω ⊂ C n → C n is called symplectic if it preserves the symplectic form ω , that is, F ∗ ω = ω . Theorem (Gromov, 1985) Let r , R > 0 . Suppose there is a symplectic embedding F : B n ( r ) → D ( R ) × C n − 1 . Then r ≤ R. What is complex here? . . . Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Gromov’s Non-Squeezing Theorem Let B n be the unit ball in C n ; then D = B 1 ⊂ C is the unit disc. B n ( r ) is the ball of radius r . Let ω = � n � n j = 1 dx j ∧ dy j = i j = 1 dz j ∧ dz j be the standard 2 symplectic form in C n = R 2 n . A smooth map F : Ω ⊂ C n → C n is called symplectic if it preserves the symplectic form ω , that is, F ∗ ω = ω . Theorem (Gromov, 1985) Let r , R > 0 . Suppose there is a symplectic embedding F : B n ( r ) → D ( R ) × C n − 1 . Then r ≤ R. What is complex here? . . . Only notation. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Gromov’s Non-Squeezing Theorem Let B n be the unit ball in C n ; then D = B 1 ⊂ C is the unit disc. B n ( r ) is the ball of radius r . Let ω = � n � n j = 1 dx j ∧ dy j = i j = 1 dz j ∧ dz j be the standard 2 symplectic form in C n = R 2 n . A smooth map F : Ω ⊂ C n → C n is called symplectic if it preserves the symplectic form ω , that is, F ∗ ω = ω . Theorem (Gromov, 1985) Let r , R > 0 . Suppose there is a symplectic embedding F : B n ( r ) → D ( R ) × C n − 1 . Then r ≤ R. What is complex here? . . . Only notation. Gromov’s proof is based on complex analysis, namely on J-complex (pseudoholomorphic) curves. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Infinite-dimensional versions Flows of Hamiltonian PDEs are symplectic transformations. Non-squeezing property is of great interest. There are many results for specific PDEs. Kuksin (1994-95) proved a general non-squeezing result for symplectomorphisms of the form F = I + compact. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Infinite-dimensional versions Flows of Hamiltonian PDEs are symplectic transformations. Non-squeezing property is of great interest. There are many results for specific PDEs. Kuksin (1994-95) proved a general non-squeezing result for symplectomorphisms of the form F = I + compact. Bourgain (1994-95) proved the result for cubic NLS. Consider time t flow F : u ( 0 ) �→ u ( t ) of the equation iu t + u xx + | u | p u = 0 , x ∈ R / Z , t > 0 . Then F is a symplectic transformation of L 2 ( 0 , 1 ) , 0 < p ≤ 2. Bourgain proved the non-squeezing property for p = 2. For other values of p the question is open. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Infinite-dimensional versions Colliander, Keel, Staffilani, Takaoka, and Tao (2005) proved the result for the KdV. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Infinite-dimensional versions Colliander, Keel, Staffilani, Takaoka, and Tao (2005) proved the result for the KdV. Roum´ egoux (2010) - BBM equation. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Infinite-dimensional versions Colliander, Keel, Staffilani, Takaoka, and Tao (2005) proved the result for the KdV. Roum´ egoux (2010) - BBM equation. Abbondandolo and Majer (2014) - in case F ( B ) is convex. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Infinite-dimensional versions Colliander, Keel, Staffilani, Takaoka, and Tao (2005) proved the result for the KdV. Roum´ egoux (2010) - BBM equation. Abbondandolo and Majer (2014) - in case F ( B ) is convex. Finally, Fabert (2015) proposes a proof of the general result using non-standard analysis. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Infinite-dimensional versions Colliander, Keel, Staffilani, Takaoka, and Tao (2005) proved the result for the KdV. Roum´ egoux (2010) - BBM equation. Abbondandolo and Majer (2014) - in case F ( B ) is convex. Finally, Fabert (2015) proposes a proof of the general result using non-standard analysis. We prove a non-squeezing result for a symplectic transformation F of the Hilbert space assuming that the derivative F ′ is bounded in Hilbert scales. We apply our result to discrete nonlinear Schr¨ odinger equations. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Hilbert scales Let H be a complex Hilbert space with fixed orthonormal basis ( e n ) ∞ n = 1 . Let ( θ n ) ∞ n = 1 be a sequence of positive numbers such that θ n → ∞ as n → ∞ , for example, θ n = n . Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Hilbert scales Let H be a complex Hilbert space with fixed orthonormal basis ( e n ) ∞ n = 1 . Let ( θ n ) ∞ n = 1 be a sequence of positive numbers such that θ n → ∞ as n → ∞ , for example, θ n = n . For s ∈ R we define H s as a Hilbert space with the following norm: � x � 2 � | x n | 2 θ 2 s � s = n , x = x n e n . Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Hilbert scales Let H be a complex Hilbert space with fixed orthonormal basis ( e n ) ∞ n = 1 . Let ( θ n ) ∞ n = 1 be a sequence of positive numbers such that θ n → ∞ as n → ∞ , for example, θ n = n . For s ∈ R we define H s as a Hilbert space with the following norm: � x � 2 � | x n | 2 θ 2 s � s = n , x = x n e n . The family ( H s ) is called the Hilbert scale corresponding to the basis ( e n ) and sequence ( θ n ) . We have H 0 = H . For s > r , the space H s is dense in H r , and the inclusion H s ⊂ H r is compact. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Hilbert scales Let H be a complex Hilbert space with fixed orthonormal basis ( e n ) ∞ n = 1 . Let ( θ n ) ∞ n = 1 be a sequence of positive numbers such that θ n → ∞ as n → ∞ , for example, θ n = n . For s ∈ R we define H s as a Hilbert space with the following norm: � x � 2 � | x n | 2 θ 2 s � s = n , x = x n e n . The family ( H s ) is called the Hilbert scale corresponding to the basis ( e n ) and sequence ( θ n ) . We have H 0 = H . For s > r , the space H s is dense in H r , and the inclusion H s ⊂ H r is compact. Example. H = L 2 ( 0 , 1 ) with the standard Fourier basis, θ n = ( 1 + n 2 ) 1 / 2 , n ∈ Z . Then H s is the standard Sobolev space. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Main Result Let B ( r ) = B ∞ ( r ) be the ball of radius r in H . Theorem Let r , R > 0 . Let F : B ( r ) → D ( R ) × H be a symplectic embedding of class C 1 . Suppose there is s 0 > 0 such that for every | s | < s 0 the derivative F ′ ( z ) is bounded in H s uniformly in z ∈ B ( r ) . Then r ≤ R. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
Discrete non-linear Schr¨ odinger equation Consider the following system of equations � n + f ( | u n | 2 ) u n + iu ′ a nk u k = 0 . (1) k Here u ( t ) = ( u n ( t )) n ∈ Z , u n ( t ) ∈ C , t ≥ 0. Alexander Tumanov Non-Squeezing for the discrete Schr¨ odinger equation
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