The Defocusing NLS Equation and Its Normal Form

Grébert, Benoît; Kappeler, Thomas

doi:10.4171/131

Cited by 93 publications

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“…Arguing as in the proof of [11,Theorem 20.3], one sees that for any n 1 and (s, p) admissible with s > −1, the frequency ω (2) n is a real analytic function of the actions on the complex neighborhood V 2s+1,p/2 of 2s+1,p/2 + introduced in (74). March 29, 2018 Proof.…”

Section: Remark 44mentioning

confidence: 87%

“…First let us recall some definitions and facts on infinite products form [11]. Let a := (a n ) n 1 be a sequence of complex numbers.…”

Section: B Infinite Productsmentioning

confidence: 99%

“…The following result is obtained from [11,Lemma C1] by considering sequences a = (a m ) m∈Z with a m = 0 for m 0. We say a sequence σ = (σ n ) n 1 of complex numbers is simple if σ m = σ n for any n = m, and define σ 0 = (n 2 π 2 ) n 1 .…”

Section: B Infinite Productsmentioning

confidence: 99%

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On the wellposedness of the KdV/KdV2 equations and their frequency maps

Kappeler¹,

Molnar²

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on H s the KdV2 equation is C 0 -wellposed if s 0 and illposed (in a strong sense) if s < 0.

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Section: Remark 44mentioning

confidence: 87%

“…First let us recall some definitions and facts on infinite products form [11]. Let a := (a n ) n 1 be a sequence of complex numbers.…”

Section: B Infinite Productsmentioning

confidence: 99%

See 1 more Smart Citation

On the wellposedness of the KdV/KdV2 equations and their frequency maps

Kappeler¹,

Molnar²

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…The KdV is a good example of an integrable PDE in the sense that properties of many other integrable equations with self-adjoint Lax operators, e.g. of the defocusing Zakharov-Shabat equation (see [25]), and of their perturbations are very similar to those of KdV and its perturbations, while the equations with nonselfadjoint Lax operators, e.g. the Sine-Gordon equation, are similar to KdV when we study their small-amplitude solutions (and the KAM-theory for such equation is similar to the KAM theory for KdV without the smallness assumption, see [44]).…”

Section: Introductionmentioning

confidence: 99%

The KdV equation under periodic boundary conditions and its perturbations

Huang¹,

Kuksin²

2014

Nonlinearity

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“…The Cauchy problem (1.1) has been studied extensively from both theoretical and applied points of view. It is known to be one of the simplest partial differential equations (PDEs) with complete integrability [1,2,19]. In the following, however, we only discuss analytical aspects of (1.1) without using the complete integrable structure of the equation.…”

mentioning

confidence: 99%

Non-Existence of Solutions for the Periodic Cubic NLS below ${L}^{{2}}$

Guo

2016

Int Math Res Notices

137

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Abstract. We prove non-existence of solutions for the cubic nonlinear Schrödinger equation (NLS) on the circle if initial data belong to H s (T) \ L 2 (T) for some s ∈ (− 1 8, 0). The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short-time Fourier restriction norm method. where u is a complex-valued function. The Cauchy problem (1.1) has been studied extensively from both theoretical and applied points of view. It is known to be one of the simplest partial differential equations (PDEs) with complete integrability [1,2,19]. In the following, however, we only discuss analytical aspects of (1.1) without using the complete integrable structure of the equation. It is well known that (1.1) enjoys several symmetries. In the following, we concentrate on the scaling symmetry and the Galilean symmetry. The scaling symmetry states that if u(x, t) is a solution to (1.1) on R with initial condition u 0 , then u λ (x, t) = λ −1 u(λ −1 x, λ −2 t) is also a solution to (1.1) with the λ-scaled initial condition u λ 0 (x) = λ −1 u 0 (λ −1 x). Associated to the scaling symmetry, there is a scaling-critical Sobolev index s c such that the homogeneousḢ sc -norm is invariant under the dilation symmetry. It is commonly conjectured that a PDE is ill-posed in H s for s < s c . In the case of the one-dimensional cubic NLS, the scaling-critical Sobolev index is s c = − 1 While there is no dilation symmetry in the periodic setting, the heuristics provided by the scaling argument plays an important role even in the periodic setting. For example, a norm inflation result for (1.1) on T analogous to [12] also holds for s < s c = − x u 0 (x). This symmetry also holds on the circle for β ∈ 2Z. Note that the L 2 -norm is invariant under the Galilean symmetry.2 This induces another critical regularity s ∞ crit = 0. Indeed, there is a strong dichotomy in the behavior of solutions for s ≥ 0 and s < 0.Bourgain [3] introduced the so-called Fourier restriction norm method via the X s,bspaces (see (3.1) below) and proved local well-posedness of (1.1) in L 2 (T). Thanks to the L 2 -conservation, this result was immediately extended to global well-posedness in L 2 (T). On the other hand, (1.1) is known to be ill-posed below L 2 (T). Burq-Gérard-Tzvetkov [5] showed that the solution map Φ t : u 0 ∈ H s → u(t) ∈ H s is not locally uniformly continuous if s < 0. See also [10]. Moreover, Christ-Colliander-Tao [11] and Molinet [31] proved that the solution map is indeed discontinuous if s < 0. Our main result in this paper states an even stronger form of ill-posedness holds true for (1.1) in negative Sobolev spaces. Theorem 1.1 (Non-existence of solutions for the cubic NLS). Let s ∈ (− 1 8 , 0) andNote that the condition (ii) is a natural condition to impose since it would follow from the continuity of the solution map :, which is one of the essential components in the well-posedness theory of evolution equations. On the one hand, the previous ill-posedness results [5,10,11,12,31] ...

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The Defocusing NLS Equation and Its Normal Form

Cited by 93 publications

References 0 publications

On the wellposedness of the KdV/KdV2 equations and their frequency maps

On the wellposedness of the KdV/KdV2 equations and their frequency maps

The KdV equation under periodic boundary conditions and its perturbations

Non-Existence of Solutions for the Periodic Cubic NLS below ${L}^{{2}}$

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