Convergence rate in Wasserstein distance and
semiclassical limit for the defocusing logarithmic Schrödinger equation

Ferriere, Guillaume

doi:10.2140/apde.2021.14.617

Cited by 12 publications

(10 citation statements)

References 25 publications

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“…The case λ < 0 (whose study of the Cauchy problem goes back to [10,22]) was recently studied by R. Carles and I. Gallagher who made explicit an unusually faster dispersion with a universal behaviour of the modulus of the solution (see [7]). The knowledge of this behaviour was very recently improved with a convergence rate but also extended through the semiclassical limit in [19]. On the other hand, the case λ > 0 seems to be the more interesting from a physical point of view and has been studied formally and rigorously (see for instance [14,24]).…”

Section: Settingmentioning

confidence: 99%

Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation

Ferriere

2021

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

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We consider the logarithmic Schrödinger equation (logNLS) in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson -a time-independent Gaussian function -is an orbitally stable solution. In this paper, we construct multi-solitons (or multi-Gaussons) for logNLS, with estimates in H 1 ∩ F (H 1 ). We also construct solutions to logNLS behaving (in L 2 ) like a sum of N Gaussian solutions with different speeds (which we call multi-gaussian). In both cases, the convergence (as t → ∞) is faster than exponential. We also prove a rigidity result on these multi-gaussians and multi-solitons, showing that they are the only ones with such a convergence.

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Section: Settingmentioning

confidence: 99%

Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation

Ferriere

2021

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

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“…This should not be too surprising, as the Fokker-Planck equation is parabolic, while we started from a Hamiltonian equation. On the other hand, in [40], by changing the strategy of proof, the convergence is improved: Denoting by W 1 the Wasserstein distance, there exists C such that…”

Section: Center Of Massmentioning

confidence: 99%

“…and it is this point of view which is adopted in [40]. The question of the strong convergence, in L 1 , of |v| 2 toward γ 2 remains open for non-Gaussian initial data.…”

Section: Center Of Massmentioning

confidence: 99%

Logarithmic Schr{ö}dinger equation and isothermal fluids

Carles

2021

Preprint

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We consider the large time behavior in two types of equations, posed on the whole space R d : the Schrödinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on the other hand. We explain some connections between the two families of equations, and show how these connections may help having an insight in all cases. We insist on some specific aspects only, and refer to the cited articles for more details, and more complete statements. We try to give a general picture of the results, and present some heuristical arguments that can help the intuition, which are not necessarily found in the mentioned articles.

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“…The case λ > 0 (whose study of the Cauchy problem goes back to [15,33]) was studied by R. Carles and I. Gallagher who made explicit an unusually faster dispersion with a universal behaviour of the modulus of the solution (see [11]). The knowledge of this behaviour was recently improved with a convergence rate but also extended through the semiclassical limit in [24]. On the other hand, the case λ < 0 seems to be the most interesting from a physical point of view and has been studied formally and rigorously (see for instance [14,19,35,13,26,25]).…”

Section: Settingmentioning

confidence: 99%

WKB analysis of the Logarithmic Nonlinear Schrodinger Equation in an analytic framework

Ferriere

2020

Preprint

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We are interested in a WKB analysis of the Logarithmic Non-Linear Schrödinger Equation with "Riemannlike" variables in an analytic framework in semiclassical regime. We show that the Cauchy problem is locally well posed uniformly in the semiclassical constant and that the semiclassical limit can be performed. In particular, our framework is not only compatible with the Gross-Pitaevskii equation with logarithmic nonlinearity, but also allows initial data (and solutions) which can converge to 0 at infinity.

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Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation

Cited by 12 publications

References 25 publications

Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation

Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation

Logarithmic Schr{ö}dinger equation and isothermal fluids

WKB analysis of the Logarithmic Nonlinear Schrodinger Equation in an analytic framework

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