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Pascal Bégout

Publications and source records attributed to Pascal Bégout.

At least 19 recordsLinked to original sources
arXiv2026

Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization

We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schr{\"o}dinger equation and dissipative parabolic dynamics through a complex timederivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails.

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arXiv2026

Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains

We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schr{\"o}dinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains.

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arXiv2026

Gradient extremals, talwegs, valleys, and directional alignment for generic gradient descent

Gradient extremals are loci along which the gradient is an eigenvector of the Hessian. These objects provide a natural geometric framework connecting several notions, notably valleys and talwegs, which we analyze from a variational viewpoint in the generic case. We then show that trajectories of the gradient flow and of its discrete counterpart exhibit directional alignment with the tangent spaces to gradient extremals, and generically to the talweg. Under non-resonance assumptions, and in contrast with the quadratic case, alignment rates are governed either by the first spectral gap or by the smallest eigenvalue of the Hessian at the limit point. Nonlinearities and the step length may both distort these rates in a complex manner. We further prove a volume concentration phenomenon emphasizing the structuring role of gradient extremals: for large times, the images of sets of initial conditions concentrate inside valleys and asymptotically around talwegs.

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arXiv2025

Solutions with expanding compact support of saturated Schr{\"o}dinger equations: self-similar solutions

We prove the existence of solutions \(u(t,x)\) of the Schr{\"o}dinger equation with a saturation nonlinear term \((u/|u|)\) having compact support, for each \(t>0,\) that expands with a growth law of the type \(C\sqrt{t}\). The primary tool is considering the self-similar solution of the associated equation. For more information see https://ejde.math.txstate.edu/Volumes/2025/53/abstr.html

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arXiv2025

On the compactness of the support of solitary waves of the complex saturated nonlinear Schr{\"o}dinger equation and related problems

We study the vectorial stationary Schr{\"o}dinger equation -$\Delta$u + a U + b u = F, with a saturated nonlinearity U = u/|u| and with some complex coefficients (a, b) $\in$ C 2 . Besides the existence and uniqueness of solutions for the Dirichlet and Neumann problems, we prove the compactness of the support of the solution, under suitable conditions on (a, b) and even when the source in the right hand side F (x) is not vanishing for large values of |x|. The proof of the compactness of the support uses a local energy method, given the impossibility of applying the maximum principle. We also consider the associate Schr{\"o}dinger-Poisson system when coupling with a simple magnetic field. Among other consequences, our results give a rigorous proof of the existence of ''solitons with compact support'' claimed, without any proof, by several previous authors.

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arXiv2024

Strong stabilization of damped nonlinear Schr{\"o}dinger equation with saturation on unbounded domains

We consider the damped nonlinear Schr\''{o}dinger equation with saturation: i.e., the complex evolution equation contains in its left hand side, besides the potential term $V(x)u,$ a nonlinear term of the form $\mathrm{i}\mu u(t,x)/|u(t,x)|$ for a given parameter $\mu >0$ (arising in optical applications on non-Kerr-like fibers). In the right hand side we assume a given forcing term $f(t,x).$ The important new difficulty, in contrast to previous results in the literature, comes from the fact that the spatial domain is assumed to be unbounded. We start by proving the existence and uniqueness of weak and strong solutions according the regularity of the data of the problem. The existence of solutions with a lower regularity is also obtained by working with a sequence of spaces verifying the Radon-Nikod\'{y}m property. Concerning the asymptotic behavior for large times we prove a strong stabilization result. For instance, in the one dimensional case we prove that there is extinction in finite time of the solutions under the mere assumption that the $L^\infty$-norm of the forcing term $f(t,x)$ becomes less than $\mu$ after a finite time. This presents some analogies with the so called feedback \textit{bang-bang controls} $v$ (here $v=-\mathrm{i}\mu u/|u|+f).$

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arXiv2024

Finite time extinction for a critically damped Schr{ö}dinger equation with a sublinear nonlinearity

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schr{ö}dinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{\varepsilon+(|u|^2)^α}u,$ with $a\in\mathbb{C},$ $\varepsilon\geqslant0,$ $2α=(1-m)$ and $m\in[0,1).$ Here we consider the sublinear case $0 0 \text{ and } 2\sqrt{m}\mathrm{Im}(z)=(1-m)\mathrm{Re}(z)\big\}.$ Among other things, we know that this damping coefficient is critical, for instance, in order to obtain the monotonicity of the associated operator (see the paper by Liskevich and Perel'muter [16] and the more recent study by Cialdea and Maz'ya [14]). The finite time extinction of solutions is proved by a suitable energy method after obtaining appropiate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.

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arXiv2024

Existence of weak solutions to some stationary Schr{ö}dinger equations with singular nonlinearity

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schr\''{o}dinger operator under the presence of a singular nonlinear term. Among other new facts, with respect some previous results in the literature for such type of nonlinear potential terms, we include the case in which the spatial domain is possibly unbounded (something which is connected with some previous localization results by the authors), the presence of possible non-local terms at the equation, the case of boundary conditions different to the Dirichlet ones and, finally, the proof of the existence of solutions when the right-hand side term of the equation is beyond the usual $L^2$-space.

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arXiv2024

Finite time extinction for a class of damped Schr{ö}dinger equations with a singular saturated nonlinearity

We present some sharper finite extinction time results for solutions of a class of damped nonlinear Schr{ö}dinger equations when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{\varepsilon+(|u|^2)^α}u,$ with $a\in\mathbb{C},$ $\varepsilon\geqslant0,$ $2α=(1-m)$ and $m\in[0,1).$ To carry out the improvement of previous results in the literature we present in this paper a careful revision of the existence and regularity of weak solutions under very general assumptions on the data. We prove that the problem can be solved in the very general framework of the maximal monotone operators theory, even under a lack of regularity of the damping term. This allows us to consider, among other things, the singular case $m=0.$ We replace the above approximation of the damping term by a different one which keeps the monotonicity for any $\varepsilon\geqslant0$. We prove that, when $m=0,$ the finite extinction time of the solution arises for merely bounded right hand side data $f(t,x).$ This is specially useful in the applications in which the Schr{ö}dinger equation is coupled with some other functions satisfying some additional equations.

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arXiv2023

The dual space of a complex Banach space restricted to the field of real numbers

Solutions of some partial differential equations are obtained as critical points of a real funtional. Then the Banach space where this functional is defined has to be real, otherwise, it is not differentiable. It follows that the equation is solved with respect to the real dual space of this Banach space. But if the solution is complex-valued there is the following problem: what does the multiplication of this equation by a complex number mean ? In this note, we explain how to rigorously define this operation.

math.AP↗
arXiv2021

Finite time extinction for the strongly damped nonlinear Schr{ö}dinger equation in bounded domains

We prove the \textit{finite time extinction property} $(u(t)\equiv 0$ on $Ω$ for any $t\ge T_\star,$ for some $T_\star>0)$ for solutions of the nonlinear Schrödinger problem ${\rm i} u_t+Δu+a|u|^{-(1-m)}u=f(t,x),$ on a bounded domain $Ω$ of $\mathbb{R}^N,$ $N\le 3,$ $a\in\mathbb{C}$ with $\Im(a)>0$ (the damping case) and under the crucial assumptions $0<m<1$ and the dominating condition $2\sqrt m\,\Im(a)\ge(1-m)|\Re(a)|.$ We use an energy method as well as several a priori estimates to prove the main conclusion. The presence of the non-Lipschitz nonlinear term in the equation introduces a lack of regularity of the solution requiring a study of the existence and uniqueness of solutions satisfying the equation in some different senses according to the regularity assumed on the data.

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arXiv2020

Finite time extinction for a damped nonlinear Schr{ö}dinger equation in the whole space

We consider a nonlinear Schr{ö}dinger equation set in the whole space with a single power of interaction and an external source. We first establish existence and uniqueness of the solutions and then show, in low space dimension, that the solutions vanish at a finite time. Under a smallness hypothesis of the initial data and some suitable additional assumptions on the external source, we also show that we can choose the upper bound on which time the solutions vanish.

math.AP↗
arXiv2018

On damped second-order gradient systems

Using small deformations of the total energy, as introduced in [31], we establish that damped second order gradient systems $$u^{\prime\prime}(t)+γu^\prime(t)+\nabla G(u(t))=0,$$may be viewed as quasi-gradient systems. In order to study the asymptotic behavior of these systems, we prove that any (nontrivial) desingularizing function appearing in KL inequality satisfies $φ(s)\ge c\sqrt s$ whenever the original function is definable and $C^2.$ Variants to this result are given. These facts are used in turn to prove that a desingularizing function of the potential $G$ also desingularizes the total energy and its deformed versions. Our approach brings forward several results interesting for their own sake: we provide an asymptotic alternative for quasi-gradient systems, either a trajectory converges, or its norm tends to infinity. The convergence rates are also analyzed by an original method based on a one-dimensional worst-case gradient system.We conclude by establishing the convergence of solutions of damped second order systems in various cases including the definable case. The real-analytic case is recovered and some results concerning convex functions are also derived.

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arXiv2015

A Generalized Interpolation Inequality and its Application to the Stabilization of Damped Equations

In this paper, we establish a generalized H{ö}lder's or interpolation inequality for weighted spaces in which the weights are non-necessarily homogeneous. We apply it to the stabilization of some damped wave-like evolution equations. This allows obtaining explicit decay rates for smooth solutions for more general classes of damping operators. In particular, for $1-d$ models, we can give an explicit decay estimate for pointwise damping mechanisms supported on any strategic point.

math.AP↗
arXiv2015

Mass Concentration Phenomena for the L^2-Critical Nonlinear Schr{ö}dinger Equation

In this paper, we show that any solution of the nonlinear Schr{ö}dinger equation $iu\_t+Δu\pm|u|^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on the Bourgain's one~\cite{MR99f:35184}, which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega~\cite{MR1671214}. We also generalize to higher dimensions the results in Keraani~\cite{MR2216444} and Merle and Vega~\cite{MR1628235}.

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arXiv2015

Self-similar solutions with compactly supported profile of some nonlinear Schr{ö}dinger equations

This paper deals with the study of "\textit{sharp localized}" solutions of a nonlinear type Schr{ö}dinger equation in the whole space $\R^N,$ $N\ge1,$ with a zero order term, in modulus, like a power $m$ less than one of the modulus of the solution, and with a non zero external forcing term $\f.$ Our fundamental assumption is that such an exponent $m$ verifies $m\in (0,1).$ The self-similar structure of the solution is justified from the assumption that the external forcing term satisfies that $\f(t,x)=t^{-(\vp-2)/2}\F(t^{-1/2}x)$ for some complex exponent $\vp$ and for some profile function $\F$ which is assumed to be with compact support in $\R^N.$ We show the existence of solutions $\vu(t,x)=t^{\vp/2}\U(t^{-1/2}x),$ with a profile $\U,$ which also have compact support in $\R^N,$ reason why we call as "\textit{sharp localized}" solutions to this type of solutions. The proof of the localization of the support of the profile $\U$ uses some suitable energy method applied to the stationary problem satisfied by $\U$ after some unknown transformation.

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