arXiv subjects

Rémi Carles

Publications and source records attributed to Rémi Carles.

At least 19 recordsLinked to original sources

arXiv2026

Soliton Dynamics for the Cubic-Quintic Nonlinear Schrödinger Equation with a Potential

We study the semiclassical dynamics of solitary waves for the three-dimensional cubic--quintic nonlinear Schrödinger equation with an external potential. For initial data given by a modulated free ground state $P_ω$, we prove that the wave-packet center follows the associated classical Hamiltonian flow. For $ω\in\mathcal I$, we obtain qualitative persistence near the soliton orbit. Under the additional positive-slope condition $ω\in\mathcal I_+$, we establish an $O(\eps)$ approximation in the scaled $H^1$ norm on every fixed time interval. We also derive $O(\eps^2)$ concentration estimates for the mass, momentum, and center of mass.

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arXiv2026

Norm inflation in negative order Sobolev spaces for KdV and KP

We prove norm inflation phenomena for KdV and KP equations in negative order Sobolev spaces, in the periodic case, as well as on the whole space, on an arbitrarily large scale of negative order Sobolev spaces as target spaces. The proof relies on WKB analysis for a semiclassical version of the equation, in a weakly nonlinear r{é}gime, and the creation of the zero Fourier mode by resonant interaction. Unlike in previous similar results, this average mode has a smaller order of magnitude than the initial data, which requires a more detailed WKB analysis.

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arXiv2026

A toy model for frequency cascade in the nonlinear Schrodinger equation

We present an elementary approach to observe frequency cascade on forced nonlinear Schr{ö}dinger equations. The forcing term (which may also appear as a potential term instead) consists of a constant term, perturbed by a modulated Gaussian well. Algebraic computations provide an explicit frequency cascade when time and space derivatives are discarded from the nonlinear Schr{ö}dinger equation. We provide stability results, showing that when derivatives are incorporated in the model, the initial algebraic solution may be little affected, possibly over long time intervals. Numerical simulations are provided, which support the analysis.

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arXiv2026

Dependence of the nonlinear Schr{ö}dinger flow upon the nonlinearity

We consider the defocusing nonlinear Schr{ö}dinger equation in the energy-subcritical case, and investigate the dependence of the solution upon the power of the nonlinearity. Special attention is paid to the global in time description. The main three aspects addressed, in the decreasing order of difficulty, are the limit when the total power tends to one, along with the connection with the logarithmic Schr{ö}dinger equation, the description when long range effects may be present, and the continuity of the scattering operator in the short range case. This text resumes the presentation given by the first author at {É}cole polytechnique for the Laurent Schwartz seminar, in May 2026.

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arXiv2026

On scattering for NLS: rigidity properties and numerical simulations via the lens transform

We analyse the scattering operator associated with the defocusing nonlinear Schr{ö}dinger equation which captures the evolution of solutions over an infinite time-interval under the nonlinear flow of this equation. The asymptotic nature of the scattering operator (involving unbounded time) makes its computation particularly challenging. We overcome this by exploiting the space-time compactification provided by the lens transform, marking the first use of this technique in numerical simulations. This results in a highly efficient and reliable methodology for computing the scattering operator in various regimes. In developing this approach we introduce and prove several new identities and theoretical properties of the scattering operator. We support our construction with several numerical experiments which we show to agree with known analytical properties of the scattering operator, and also address the case of long-range scattering for the one-dimensional cubic Schr{ö}dinger equation. Our simulations permit us to further explore regimes beyond current analytical understanding, and lead us to formulate new conjectures concerning fixed and rotating points of the operator, as well as its existence in the long-range setting for both defocusing and focusing cases.

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arXiv2026

On the ground state of the nonlinear Schr{ö}dinger equation: asymptotic behavior at the endpoint powers

We consider the ground states of the nonlinear Schr{ö}dinger equation, which stand for radially symmetric and exponentially decaying solutions on the full space. We investigate their behaviors at both endpoint powers of the nonlinearity, up to some rescaling to infer non-trivial limits. One case corresponds to the limit towards a Gaussian function called Gausson, which is the ground state of the stationary logarithmic Schr{ö}dinger equation. The other case, for dimension at least three, corresponds to the limit towards the Aubin-Talenti algebraic soliton. We prove strong convergence with explicit bounds for both cases, and provide detailed asymptotics. These theoretical results are illustrated with numerical approximations.

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arXiv2025

Small-time approximate controllability of the logarithmic Schr\''dinger equation

We consider Schr{ö}dinger equations with logarithmic nonlinearity and bilinear controls, posed on $\mathbb{T}^d$ or $\mathbb{R}^d$. We prove their small-time global $L^2$-approximate controllability. The proof consists in extending to this nonlinear framework the approach introduced by the first and third authors in \cite{beauchard-pozzoli2} to control the linear equation: it combines the small-time controllability of phases and gradient flows. Due to the nonlinearity, the required estimates are more difficult to establish than in the linear case. The proof here is inspired by WKB analysis. This is the first result of (small-time) global approximate controllability, for nonlinear Schr{ö}dinger equations, with bilinear controls.

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arXiv2025

On the dependence of the nonlinear Schrodinger flow upon the power of the nonlinearity

We prove continuity properties for the flow map associated to the defocusing energy-subcritical power-like nonlinear Schr{ö}dinger equation, when the power varies. We show local in time continuity in the energy space for any power, and global in time continuity for sufficiently large powers. When the linear dispersive rate is counterbalanced by a time-dependent rescaling, we show a uniform in time continuity of the squared modulus of this rescaled function, in Kantorovich distance, for any power, including long range cases in terms of scattering. The most difficult result addresses the convergence of suitably renormalized solutions to the solution of the logarithmic Schr{ö}dinger equation, when the power goes to zero, uniformly in time, in Kantorovich distance. The proof relies on estimates for perturbed porous medium equations, involving the harmonic Fokker-Planck operator.

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arXiv2025

Time splitting and error estimates for nonlinear Schrodinger equations with a potential

We consider the nonlinear Schr{ö}dinger equation with a potential, also known as Gross-Pitaevskii equation. By introducing a suitable spectral localization, we prove low regularity error estimates for the time discretization corresponding to an adapted Lie-Trotter splitting scheme. The proof is based on tools from spectral theory and pseudodifferential calculus in order to obtain various estimates on the spectral localization, including discrete Strichartz estimates which support the nonlinear analysis.

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arXiv2025

Propagation of coherent states in the logarithmic Schrodinger equation

We consider the logarithmic Schr{ö}dinger equation in a semiclassical scaling, in the presence of a smooth, at most quadratic, external potential. For initial data under the form of a single coherent state, we identify the notion of criticality as far as the nonlinear coupling constant is concerned, in the semiclassical limit. In the critical case, we prove a general error estimate, and improve it in the case of initial Gaussian profiles. In this critical case, when the initial datum is the sum of two Gaussian coherent states with different centers in phase space, we prove a nonlinear superposition principle.

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arXiv2024

On the Cauchy problem for logarithmic fractional Schr{ö}dinger equation

We consider the fractional Schrodinger equation with a logarithmic nonlinearity, when the power of the Laplacian is between zero and one. We prove global existence results in three different functional spaces: the Sobolev space corresponding to the quadratic form domain of the fractional Laplacian, the energy space, and a space contained in the operator domain of the fractional Laplacian. For this last case, a finite momentum assumption is made, and the key step consists in estimating the Lie commutator between the fractional Laplacian and the multiplication by a monomial.

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arXiv2024

Logarithmic Gross-Pitaevskii equation

We consider the Schr{ö}dinger equation with a logarithmic nonlinearty and non-trivial boundary conditions at infinity. We prove that the Cauchy problem is globally well posed in the energy space, which turns out to correspond to the energy space for the standard Gross-Pitaevskii equation with a cubic nonlinearity, in small dimensions. We then characterize the solitary and travelling waves in the one dimensional case.

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arXiv2023

Low regularity solutions to the logarithmic Schrodinger equation

We consider the logarithmic Schr{ö}dinger equation, in various geometric settings. We show that the flow map can be uniquely extended from H^1 to L^2 , and that this extension is Lipschitz continuous. Moreover, we prove the regularity of the flow map in intermediate Sobolev spaces.

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arXiv2023

Pathological set with loss of regularity for nonlinear Schr{ö}dinger equations

We consider the mass-supercritical, defocusing, nonlinear Schr{ö}dinger equation. We prove loss of regularity in arbitrarily short times for regularized initial data belonging to a dense set of any fixed Sobolev space for which the nonlinearity is supercritical. The proof relies on the construction of initial data as a superposition of disjoint bubbles at different scales. We get an approximate solution with a time of existence bounded from below, provided by the compressible Euler equation, which enjoys zero speed of propagation. Introducing suitable renormalized modulated energy functionals, we prove spatially localized estimates which make it possible to obtain the loss of regularity.

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arXiv2022

Scattering and uniform in time error estimates for splitting method in NLS

We consider the nonlinear Schr{ö}dinger equation with a defocusing nonlinearity which is mass-(super)critical and energy-subcritical. We prove uniform in time error estimates for the Lie-Trotter time splitting discretization. This uniformity in time is obtained thanks to a vectorfield which provides time decay estimates for the exact and numerical solutions. This vectorfield is classical in scattering theory, and requires several technical modifications compared to previous error estimates for splitting methods.

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arXiv2022

On ground states for the 2D Schrodinger equation with combined nonlinearities and harmonic potential

We consider the nonlinear Schr{ö}dinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a framework includes physical models, and ensures that finite energy solutions are global in time. We address the questions of the existence and the orbital stability of the set of standing waves. Given the mathematical features of the equation (external potential and inhomogeneous nonlinearity), the set of parameters for which standing waves exist in unclear. In the twodimensional case, we adapt the method of fundamental frequency solutions, introduced by the second author in the higher dimensional case without potential. This makes it possible to describe accurately the set of fundamental frequency standing waves and ground states, and to prove its orbital stability.

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arXiv2022

On the Cauchy problem for the Hartree approximation in quantum dynamics

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schr{ö}dinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the the time-dependent Dirac-Frenkel variational principle. We handle the case of Coulomb potentials thanks to Strichartz estimates. Our main result addresses a general setting where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations.

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arXiv2022

Logarithmic Schr{ö}dinger equation and isothermal fluids

We consider the large time behavior in two types of equations, posed on the whole space R^d: the Schr{ö}dinger 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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