Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation

Dodson, Benjamin; Lührmann, Jonas; Mendelson, Dana

doi:10.1016/j.aim.2019.02.001

Cited by 54 publications

(73 citation statements)

References 72 publications

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“…The inequality (19) can be found in [27,Lemma 5.10]. An inequality similar to (20) can be found in [27,Lemma 5.11], and we present a different argument. Using duality and pP N χ 1 P M q˚" P M χ 1 P N , we can assume that N ě M .…”

Section: Harmonic Analysismentioning

confidence: 50%

“…In this paper, we are mainly concerned with the case M " 1. Then, (27) describes the linear evolution on short time intervals more accurately than (28). As a corollary of the refined dispersive estimate, we obtain the following refined Strichartz estimate.…”

Section: Strichartz Estimatesmentioning

confidence: 81%

“…The method of [26] has since been used in several related works. In [27], Dodson, Lührmann, and Mendelson used local energy decay to improve the regularity condition to s ą 0. After replacing the cubes in the Wiener randomization by thin annuli, the author proved almost sure scattering for radial data in dimension three [13].…”

Section: Figure 1: Partions Of Phase Spacementioning

confidence: 99%

“…The main new ingredient is an interaction flux estimate between the linear and nonlinear components of the solution. Finally, the almost sure scattering for the radial energy-critical nonlinear Schrödinger equation in four dimensions has been obtained in [27,33].…”

Section: Figure 1: Partions Of Phase Spacementioning

confidence: 99%

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Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Bringmann

2020

International Mathematics Research Notices

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We study the defocusing energy-critical nonlinear wave equation in four dimensions. Our main result proves the stability of the scattering mechanism under random pertubations of the initial data. The random pertubation is defined through a microlocal randomization, which is based on a unit-scale decomposition in physical and frequency space. In contrast to the previous literature, we do not require the spherical symmetry of the pertubation. The main novelty lies in a wave packet decomposition of the random linear evolution. Through this decomposition, we can adaptively estimate the interaction between the rough and regular components of the solution. Our argument relies on techniques from restriction theory, such as Bourgain's bush argument and Wolff's induction on scales.Contents MSC2010 : 35L05, 42B20, 42B37

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Section: Harmonic Analysismentioning

confidence: 50%

Section: Strichartz Estimatesmentioning

confidence: 81%

Section: Figure 1: Partions Of Phase Spacementioning

confidence: 99%

Section: Figure 1: Partions Of Phase Spacementioning

confidence: 99%

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Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Bringmann

2020

International Mathematics Research Notices

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“…A key difference in this setting, however, is that we will first need to treat an equation for the projection of the solution onto the negative eigenvalue. As in [24,25], we will work with the framework of a generalized forced nonlinear wave equation, but in this case we will only use this framework for the component of the solution projected onto the absolutely continuous subspace of the linearized operator H a , see Section 7 as well as the system of equations (7.8) -(7.10) for more details.…”

Section: Satisfiesmentioning

confidence: 99%

The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

Kenig

Mendelson

2019

International Mathematics Research Notices

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We consider the focusing energy-critical quintic nonlinear wave equation in three dimensional Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve inḢ s, for any s > 1/2. By randomizing radial initial data inḢ sfor s > 5/6, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton which give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the first long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.1.1. Statement of the main theorem and discussion of methods. We consider the stationary solutions φ a and we make the following ansatz:We take the scaling parameter a(t) to be time dependent, and as remarked in [40], this leads to analysis similar to modulation theory, but distinct since resonances are not associated with L 2 projections. The resonance ϕ a(t) := ∂ a φ a(t)from the definition of P ac , see (3.3) below, we see that the condition in the X s norm is weaker than requiring the same bounds for the function f itself. Moreover, this definition highlights the fact that the Lorentz condition will only be required for the low-frequency component, see (7.4).Let s ∈ R and definefor some ε < ε 0 , where ε 0 , C, c > 0 are determined in Theorem 1.5 below.Remark 1.3. In the case of finite energy initial data, the orthogonality conditionensures that the second variation of the energy restricted to a certain codimension one Lipschitz submanifold is non-negative at φ a , see the discussion following [40, Theorem 1]. Although this does not apply to our (infinite energy) initial data, more concretely, we use this condition to avoid growth when solving an ODE for the pure point component of the solution, see (7.12).

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Large deviations principle for the cubic NLS equation

Garrido

Grande

Kurianski

et al. 2023

Comm Pure Appl Math

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In this paper, we present a probabilistic study of rare phenomena of the cubic nonlinear Schrödinger equation on the torus in a weakly nonlinear setting. This equation has been used as a model to numerically study the formation of rogue waves in deep sea. Our results are twofold: first, we introduce a notion of criticality and prove a Large Deviations Principle (LDP) for the subcritical and critical cases. Second, we study the most likely initial conditions that lead to the formation of a rogue wave, from a theoretical and numerical point of view. Finally, we propose several open questions for future research.

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Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation

Cited by 54 publications

References 72 publications

Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

Large deviations principle for the cubic NLS equation

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