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Your search keyword '"Duboscq, Romain"' showing total 5 results
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5 results on '"Duboscq, Romain"'

1. Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library

Author

Besse, Christophe, Duboscq, Romain, Le Coz, Stefan, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Numerical Analysis, FOS: Physical sciences, Numerical Analysis (math.NA), Pattern Formation and Solitons (nlin.PS), Nonlinear Sciences - Pattern Formation and Solitons, Ground states, Computational Mathematics, Mathematics - Analysis of PDEs, [NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], Quantum Graphs, Modeling and Simulation, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Finite Differences, Mathematics - Numerical Analysis, Python Library, Nonlinear Schrodinger equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

International audience; Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schrödinger equation. Whereas in the last ten years they have known considerable developments on the theoretical side, their study from the numerical point of view remains in its early stages. The goal of this paper is to present the Grafidi library, a Python library which has been developed with the numerical simulation of nonlinear Schrödinger equations on graphs in mind. We will show how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. We will also simulate the dynamics of the nonlinear Schrödinger equation with a Crank-Nicolson relaxation scheme and a Strang splitting scheme. Finally, in a series of numerical experiments on various types of graphs, we will compare the outcome of our numerical calculations for ground states with the existing theoretical results, thereby illustrating the versatility and efficiency of our implementations in the framework of the Grafidi library.

Published
2021
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2. Gradient Flow Approach to the Calculation of Ground States on Nonlinear Quantum Graphs

Author

Besse, Christophe, Duboscq, Romain, Coz, Stefan Le, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-11-IDEX-0002,UNITI,Institut for Advanced Study in Toulouse(2011), ANR-14-CE25-0009,MAToS,Analyse des singularités topologiques dans quelques problèmes issus de la physique mathématique(2014), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), ANR-11-IDEX-0002,UNITI,Université Fédérale de Toulouse(2011), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ocean Engineering, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

International audience; We introduce and implement a method to compute stationary states of nonlinear Schr\"odinger equations on metric graphs. Stationary states are obtained as local minimizers of the nonlinear Schr\"odinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) towards stationary states. We then establish the link between the continuous flow and its discretized version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute stationary states. Further experiments as well as detailed explanation of our numerical algorithm are given in a companion paper.

Published
2020
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3. On a stochastic Hardy-Littlewood-Sobolev inequality with application to Strichartz estimates for a noisy dispersion

Author

Duboscq , Romain, Réveillac , Anthony, Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), and Institut National des Sciences Appliquées ( INSA )

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Non-linear Schrödinger equation, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Stochastic regularization, Stochastic Partial Differential Equations, Hardy-Littlewood-Sobolev inequality, Nonlinear Schrödinger equation, Stochastic Partial Dierential Equations, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], AMS 2010 subject classification: Primary: 35Q55, Secondary: 60H15, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Abstract

In this paper, we investigate a stochastic Hardy-Littlewood-Sobolev inequality. Due to the stochastic nature of the inequality, the relation between the exponents of inte-grability is modied. This modication can be understood as a regularization by noise phenomenon. As a direct application, we derive Strichartz estimates for the white noise dispersion which enables us to address a conjecture from [3]. Let (Ω, P) be the standard probability space endowed with the Wiener ltration (F t) t≥0. The main objective of this paper is to address the local Cauchy problem for the following nonlinear Schrödinger equation with white noise dispersion idψ(t, x) = −∆ψ(t, x) • dW t (ω) + λ|ψ| 2σ ψ(t, x)dt, ∀(t, x, ω) ∈ [0, T ] × R d × Ω, ψ(0, ·) = ψ 0 ∈ L 2 (R d), (1.1) where λ ∈ R and (W t) t≥0 is the Wiener process and the product • is understood in the Stratonovich sense. For d = 1 and σ = 1, this equation arises in the eld of nonlinear optics as a limit model for the propagation of light pulse in an optical ber where the dispersion varies along the ber [2, 1]. These variations in the dispersion accounts for the so-called dispersion management which aims to improve the transmission of a light signal by constructing a zero-mean dispersion ber in order to avoid the problem of the chromatic dispersion of the light signal. When the variations are assumed to be random, a noisy dispersion can be derived (see [24, 12]) which leads, in the white noise case, to Equation

Published
2019

4. On Strichartz estimates for a dispersion modulated by a time-dependent deterministic noise

Author

Duboscq , Romain, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Centre National de la Recherche Scientifique ( CNRS ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -PRES Université de Toulouse-Université Paul Sabatier - Toulouse 3 ( UPS ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse 1 Capitole ( UT1 ), Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), and Institut National des Sciences Appliquées (INSA)

Subjects
Secondary: 60H15, Regularization by noise AMS 2010 subject classification: Primary: 35Q55, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Hardy-Littlewood-Sobolev inequality, Nonlinear Schrödinger equation, Mathematics::Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Strichartz estimate, Analysis of PDEs (math.AP)
Abstract

We address the Cauchy problem for a nonlinear Schr{\"o}dinger equation where the dispersion is modulated by a deterministic noise. The noise is understood as the derivative of a self-affine function of order H $\in$ (0, 1). Due to the self-similarity of the noise, we obtain modified Strichartz estimates which enables us to prove the global well-posedness of the equation for L2-supercritical nonlinearities. This is an occurence of regularization by noise in a purely deterministic context.

Published
2018
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5. Vortex patterns in the almost-bosonic anyon gas

Author

Douglas Lundholm, Romain Duboscq, Michele Correggi, Nicolas Rougerie, Dipartimento di Matematica 'Guido Castelnuovo' [Roma I] (Sapienza University of Rome), Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome], Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA), Department of Mathematics [Sweden] (KTH), Stockholm University, Laboratoire de physique et modélisation des milieux condensés (LPM2C), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), European Project: 758620,CORFRONMAT, Correggi, Michele, Duboscq, Romain, Lundholm, Dougla, Rougerie, Nicolas, Dipartimento di Matematica 'Guido Castelnuovo' [Roma I] ( Sapienza University of Rome ), Università degli Studi di Roma 'La Sapienza' [Rome], Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ), Department of Mathematics [Sweden] ( KTH ), Laboratoire de physique et modélisation des milieux condensés ( LPM2C ), Université Joseph Fourier - Grenoble 1 ( UJF ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), European Project : 758620,CORFRONMAT, Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome] (UNIROMA), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT), and European Project: 758620,H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC),CORFRONMAT(2018)

Subjects
[ PHYS.QPHY ] Physics [physics]/Quantum Physics [quant-ph], [PHYS.COND.GAS]Physics [physics]/Condensed Matter [cond-mat]/Quantum Gases [cond-mat.quant-gas], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Anyon, General Physics and Astronomy, FOS: Physical sciences, Trapping, 01 natural sciences, Topological quantum computer, [ PHYS.COND.GAS ] Physics [physics]/Condensed Matter [cond-mat]/Quantum Gases [cond-mat.quant-gas], 010305 fluids & plasmas, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Quantum mechanics, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Abelian group, 010306 general physics, Settore MAT/07 - Fisica Matematica, Mathematical Physics, Energy functional, Physics, Condensed Matter::Quantum Gases, Quantum Physics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Mathematical Physics (math-ph), Numerical Analysis (math.NA), Gauge (firearms), Vortex, Quantum Gases (cond-mat.quant-gas), [ PHYS.MPHY ] Physics [physics]/Mathematical Physics [math-ph], Ground state, Quantum Physics (quant-ph), Condensed Matter - Quantum Gases, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

International audience; We study theoretically and numerically the ground state of a gas of 2D abelian anyons in an external trapping potential. We treat anyon statistics in the magnetic gauge picture, perturbatively around the bosonic end. This leads to a mean-field energy functional, whose ground state displays vortex lattices similar to those found in rotating Bose-Einstein condensates. A crucial difference is however that the vortex density is proportional to the underlying matter density of the gas.

Published
2019

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