Displaying similar documents to “Quasi-periodic solutions with Sobolev regularity of NLS on 𝕋 d with a multiplicative potential”

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia, Vadim Kaloshin (2015)

Journal of the European Mathematical Society

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We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix s > 1 . Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with s -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is c > 0 such that for any 𝒦 1 we find a solution u and a time T such that u ( T ) H s 𝒦 u ( 0 ) H s . Moreover, the time T satisfies the polynomial bound 0 < T < 𝒦 C .

Recent results on KAM for multidimensional PDEs

Benoît Grébert (2014)

Journées Équations aux dérivées partielles

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In this short overview I present some recent results about the KAM theory for multidimensional partial differential equations (PDEs) trying to avoid technicalities. In particular I will not state a precise KAM theorem but I will focus on the dynamical consequences for the PDEs: the existence and the stability (or not) of quasi periodic in time solutions. Concretely, I present the complete study of the nonlinear beam equation on the d -dimensional torus recently obtained in collaboration...

Waves in Honeycomb Structures

Charles L. Fefferman, Michael I. Weinstein (2012)

Journées Équations aux dérivées partielles

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We review recent work of the authors on the non-relativistic Schrödinger equation with a honeycomb lattice potential, V . In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of H V = - Δ + V and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution e - i H V t ψ 0 , for data ψ 0 , which is spectrally localized near a Dirac point. We conclude with a formal derivation and discussion...

On the number of positive solutions of singularly perturbed 1D nonlinear Schrödinger equations

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka (2006)

Journal of the European Mathematical Society

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We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When V ( x ) has multiple critical points, (1.1) has a wide variety of positive solutions for small ε and the number of positive solutions increases to as ε 0 . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of V ( x ) . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

Symmetries of the nonlinear Schrödinger equation

Benoît Grébert, Thomas Kappeler (2002)

Bulletin de la Société Mathématique de France

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Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum &lt; λ k - λ k + &lt; λ k + 1 - of a Zakharov-Shabat operator is symmetric,. λ k ± = - λ - k for all k , if and only if the sequence ( γ k ) k of gap lengths, γ k : = λ k + - λ k - , is symmetric with respect to k = 0 .

Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Antonio Ambrosetti, Veronica Felli, Andrea Malchiodi (2005)

Journal of the European Mathematical Society

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We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V ( x ) | x | α , 0 < α < 2 , and K ( x ) | x | β , β > 0 . Working in weighted Sobolev spaces, the existence of ground states v ε belonging to W 1 , 2 ( N ) is proved under the assumption that σ < p < ( N + 2 ) / ( N 2 ) for some σ = σ N , α , β . Furthermore, it is shown that v ε are spikes concentrating at a minimum point of 𝒜 = V θ K 2 / ( p 1 ) , where θ = ( p + 1 ) / ( p 1 ) 1 / 2 .

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in 3

M. Burak Erdoğan, Michael Goldberg, Wilhelm Schlag (2008)

Journal of the European Mathematical Society

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We present a novel approach for bounding the resolvent of H = - Δ + i ( A · + · A ) + V = : - Δ + L 1 for large energies. It is shown here that there exist a large integer m and a large number λ 0 so that relative to the usual weighted L 2 -norm, ( L ( - Δ + ( λ + i 0 ) ) - 1 ) m < 1 2 2 for all λ > λ 0 . This requires suitable decay and smoothness conditions on A , V . The estimate (2) is trivial when A = 0 , but difficult for large A since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

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We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), x N , u H ¹ ( N ) , where λ is a positive parameter, a and b are positive functions, while f : is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes

Jacek Dziubański, Jacek Zienkiewicz (2003)

Colloquium Mathematicae

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Let A = -Δ + V be a Schrödinger operator on d , d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of H A p if the maximal function s u p t > 0 | T t f ( x ) | belongs to L p ( d ) , where T t t > 0 is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space H A p admits a special atomic decomposition.

Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

Andrea R. Nahmod, Gigliola Staffilani (2015)

Journal of the European Mathematical Society

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We also prove a long time existence result; more precisely we prove that for fixed T > 0 there exists a set Σ T , ( Σ T ) > 0 such that any data φ ω ( x ) H γ ( 𝕋 3 ) , γ < 1 , ω Σ T , evolves up to time T into a solution u ( t ) with u ( t ) - e i t Δ φ ω C ( [ 0 , T ] ; H s ( 𝕋 3 ) ) , s = s ( γ ) > 1 . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space H 1 ( 𝕋 3 ) , that is in the supercritical scaling regime.

On the equivalence of Green functions for general Schrödinger operators on a half-space

Abdoul Ifra, Lotfi Riahi (2004)

Annales Polonici Mathematici

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We consider the general Schrödinger operator L = d i v ( A ( x ) x ) - μ on a half-space in ℝⁿ, n ≥ 3. We prove that the L-Green function G exists and is comparable to the Laplace-Green function G Δ provided that μ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schrödinger operators with potentials in the Kato class at infinity K considered by Zhao and Pinchover. As an application we study the cone L ( ) of all positive L-solutions continuously...

Optimal potentials for Schrödinger operators

Giuseppe Buttazzo, Augusto Gerolin, Berardo Ruffini, Bozhidar Velichkov (2014)

Journal de l’École polytechnique — Mathématiques

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We consider the Schrödinger operator - Δ + V ( x ) on H 0 1 ( Ω ) , where Ω is a given domain of d . Our goal is to study some optimization problems where an optimal potential V 0 has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

Periodic solutions of Euler-Lagrange equations with sublinear potentials in an Orlicz-Sobolev space setting

Sonia Acinas, Fernando Mazzone (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain existence results of periodic solutions of hamiltonian systems in the Orlicz-Sobolev space W 1 L Φ ( [ 0 , T ] ) . We employ the direct method of calculus of variations and we consider  a potential  function F satisfying the inequality | F ( t , x ) | b 1 ( t ) Φ 0 ' ( | x | ) + b 2 ( t ) , with b 1 , b 2 L 1 and  certain N -functions Φ 0 .

On the uniqueness of periodic decomposition

Viktor Harangi (2011)

Fundamenta Mathematicae

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Let a , . . . , a k be arbitrary nonzero real numbers. An ( a , . . . , a k ) -decomposition of a function f:ℝ → ℝ is a sum f + + f k = f where f i : is an a i -periodic function. Such a decomposition is not unique because there are several solutions of the equation h + + h k = 0 with h i : a i -periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the ( a , . . . , a k ) -decomposition is essentially unique. We characterize those periods for which essential...

Hardy spaces H¹ for Schrödinger operators with certain potentials

Jacek Dziubański, Jacek Zienkiewicz (2004)

Studia Mathematica

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Let K t t > 0 be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to H ¹ L if | | s u p t > 0 | K t f ( x ) | | | L ¹ ( d x ) < . We state conditions on V and K t which allow us to give an atomic characterization of the space H ¹ L .

Existence and uniqueness of periodic solutions for odd-order ordinary differential equations

Yongxiang Li, He Yang (2011)

Annales Polonici Mathematici

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The paper deals with the existence and uniqueness of 2π-periodic solutions for the odd-order ordinary differential equation u ( 2 n + 1 ) = f ( t , u , u ' , . . . , u ( 2 n ) ) , where f : × 2 n + 1 is continuous and 2π-periodic with respect to t. Some new conditions on the nonlinearity f ( t , x , x , . . . , x 2 n ) to guarantee the existence and uniqueness are presented. These conditions extend and improve the ones presented by Cong [Appl. Math. Lett. 17 (2004), 727-732].

A radial estimate for the maximal operator associated with the free Schrödinger equation

Sichun Wang (2006)

Studia Mathematica

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Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator S d and its associated global maximal operator S * * d by ( S d f ) ( x , t ) = 1 / ( 2 π ) e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ , f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, ( S * * d f ) ( x ) = s u p t | 1 / ( 2 π ) e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ | , f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, S d f is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥...

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica Musso, Frank Pacard, Juncheng Wei (2012)

Journal of the European Mathematical Society

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We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations Δ u - u + f ( u ) = 0 in N , u H 1 ( N ) , where N 2 . Under natural conditions on the nonlinearity f , we prove the existence of 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 in any dimension N 2 . Our result complements earlier works of Bartsch and Willem ( N = 4 𝚘𝚛 N 6 ) and Lorca-Ubilla ( N = 5 ) where solutions invariant under the action of O ( 2 ) × O ( N - 2 ) are constructed. In contrast, the solutions we construct are invariant under the action of D k × O ( N - 2 ) where D k O ( 2 ) denotes the dihedral...

Boundedness of sublinear operators in Triebel-Lizorkin spaces via atoms

Liguang Liu, Dachun Yang (2009)

Studia Mathematica

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Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space p , q s ( ) to a quasi-Banach space ℬ if and only if sup | | T ( a ) | | : a is an infinitely differentiable (p,q,s)-atom of p , q s ( ) < ∞, where the (p,q,s)-atom of p , q s ( ) is as defined by Han, Paluszyński and Weiss.

Control for Schrödinger operators on 2-tori: rough potentials

Jean Bourgain, Nicolas Burq, Maciej Zworski (2013)

Journal of the European Mathematical Society

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For the Schrödinger equation, ( i t + ) u = 0 on a torus, an arbitrary non-empty open set Ω provides control and observability of the solution: u t = 0 L 2 ( 𝕋 2 ) K T u L 2 ( [ 0 , T ] × Ω ) . We show that the same result remains true for ( i t + - V ) u = 0 where V L 2 ( 𝕋 2 ) , and 𝕋 2 is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for V C ( 𝕋 2 ) and conjectured for V L ( 𝕋 2 ) . The higher dimensional generalization remains open for V L ( 𝕋 n ) .

On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

Xuwen Chen, Justin Holmer (2016)

Journal of the European Mathematical Society

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We consider the 3D quantum BBGKY hierarchy which corresponds to the N -particle Schrödinger equation. We assume the pair interaction is N 3 β 1 V ( B β ) . For the interaction parameter β ( 0 , 2 / 3 ) , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the N limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for β ( 0 , 3 / 5 ) in [28]. This allows, in the case β ( 0 , 3 / 5 ) , for the application of the Klainerman–Machedon...

C 1 self-maps on closed manifolds with finitely many periodic points all of them hyperbolic

Jaume Llibre, Víctor F. Sirvent (2016)

Mathematica Bohemica

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Let X be a connected closed manifold and f a self-map on X . We say that f is almost quasi-unipotent if every eigenvalue λ of the map f * k (the induced map on the k -th homology group of X ) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of λ as eigenvalue of all the maps f * k with k odd is equal to the sum of the multiplicities of λ as eigenvalue of all the maps f * k with k even. We prove that if f is C 1 having finitely many periodic points all of them...

Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

Anne-Laure Dalibard (2011)

Journal of the European Mathematical Society

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This article investigates the long-time behaviour of parabolic scalar conservation laws of the type t u + div y A ( y , u ) - Δ y u = 0 , where y N and the flux A is periodic in y . More specifically, we consider the case when the initial data is an L 1 disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in L 1 norm like a self-similar profile for large times. The proof uses a time and space change of variables...

The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian

Jean Mawhin (2006)

Journal of the European Mathematical Society

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We prove an Ambrosetti–Prodi type result for the periodic solutions of the equation ( | u ' | p 2 u ' ) ) ' + f ( u ) u ' + g ( x , u ) = t , when f is arbitrary and g ( x , u ) + or g ( x , u ) when | u | . The proof uses upper and lower solutions and the Leray–Schauder degree.