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Guàrdia, Marcel; Haus, Emanuele; Procesi, Michela

doi:10.1016/j.aim.2016.06.018

Cited by 51 publications

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“…Evolution problems with nonlocal dispersion such as (1.1) naturally arise in various physical settings, including continuum limits of lattice systems [25], models for wave turbulence [6,30], and gravitational collapse [10,12]. The phenomenon that we study in this paper is the growth of high Sobolev norms in infinite dimensional Hamiltonian systems, which has attracted considerable attention over the past twenty years [2,49,30,4,52,6,7,13,43,21,19,22,23,20,17] . The aim of this paper is to develop a robust approach for constructing solutions whose high Sobolev norms grow over time, based on multisolitary wave interactions.…”

Section: Introductionmentioning

confidence: 99%

A Two-Soliton with Transient Turbulent Regime for the Cubic Half-Wave Equation on the Real Line

et al. 2018

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We consider the focusing cubic half-wave equation on the real lineWe construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small L 2 -norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its H 1 -norm on a finite time interval, followed by (ii) a saturation regime in which the H 1 -norm remains stationary large forever in time.

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Section: Introductionmentioning

confidence: 99%

A Two-Soliton with Transient Turbulent Regime for the Cubic Half-Wave Equation on the Real Line

et al. 2018

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“…In [37], the existence of unbounded, in H s with s > 1, solutions to the cubic defocusing Schrödinger equation was established for the first time, in the case where the equation is considered on the domain R × T d , d 2: for some solutions, the Sobolev norms grow logarithmically along a sequence of times. See also [17,33,32], in the space periodic case. For other equations (cubic Szegő equation or half-wave equation), with specific initial data, a growth rate can be exhibited, possibly along a sequence of time; see [27,28,52].…”

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confidence: 99%

Universal dynamics for the defocusing logarithmic Schrödinger equation

Carles¹,

Gallagher²

2018

Duke Math. J.

140

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We consider the nonlinear Schrödinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data, and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in working in hydrodynamical variables to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker-Planck operator.

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“…Guardia and Kaloshin revisited the proof in [18] and obtained quantitative upper bounds on the time T required for the growth to happen. Those results were generalized to the higher-order analytic nonlinearities in [29,19].…”

Section: Recent Resultsmentioning

confidence: 99%

Out-of-equilibrium dynamics and statistics of dispersive PDE

Hani¹

2017

Journées Équations Aux Dérivées Partielles

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The purpose of this note is to report on some recent advances in the study of out-ofequilibrium behavior of dispersive PDE. One can address this problematic from two different perspectives: a dynamical systems one, and a statistical physics one. The dynamical systems perspective corresponds to constructing solutions exhibiting "energy cascade" between scales, whereas the statistical physics perspective corresponds to deriving effective equations for the dynamics under some "macroscopic limits" in what is often called wave turbulence theory. The rigorous justification of this theory is an outstanding open problem from a rigorous mathematical point of view, and we will touch on it here. We shall discuss some recent attempts to better understand both of the above perspectives.

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Growth of Sobolev norms for the analytic NLS onT2

Cited by 51 publications

References 50 publications

A Two-Soliton with Transient Turbulent Regime for the Cubic Half-Wave Equation on the Real Line

A Two-Soliton with Transient Turbulent Regime for the Cubic Half-Wave Equation on the Real Line

Universal dynamics for the defocusing logarithmic Schrödinger equation

Out-of-equilibrium dynamics and statistics of dispersive PDE

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