The Experts below are selected from a list of 45945 Experts worldwide ranked by ideXlab platform
Benoît Grébert - One of the best experts on this subject based on the ideXlab platform.
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Hamiltonian Interpolation of Splitting Approximations for Nonlinear PDEs
Foundations of Computational Mathematics, 2011Co-Authors: Erwan Faou, Benoît GrébertAbstract:We consider a wide class of semilinear Hamiltonian partial differential Equations and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical trajectory remains at least uniformly integrable with respect to an eigenbasis of the linear operator (typically the Fourier basis). We show the existence of a modified interpolated Hamiltonian Equation whose exact solution coincides with the discrete flow at each time step over a long time. While for standard splitting or implicit–explicit schemes, this long time depends on a cut-off condition in the high frequencies (CFL condition), we show that it can be made exponentially large with respect to the step size for a class of modified splitting schemes.
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Hamiltonian interpolation of splitting approximations for nonlinear PDEs
Foundations of Computational Mathematics, 2011Co-Authors: Erwan Faou, Benoît GrébertAbstract:We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni- formly integrable with respect to an eigenbasis of the linear operator (typically the Fourier basis). We show the existence of a modified interpolated Hamiltonian Equation whose exact solution coincides with the discrete flow at each time step over a long time depending on a non resonance condition satisfied by the stepsize. We introduce a class of modified splitting schemes fulfilling this condition at a high order and prove for them that the numerical flow and the continuous flow remain close over exponentially long time with respect to the step size. For stan- dard splitting or implicit-explicit scheme, such a backward error analysis result holds true on a time depending on a cut-off condition in the high frequencies (CFL condition). This analysis is valid in the case where the linear operator has a discrete (bounded domain) or continuous (the whole space) spectrum.
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NORMAL FORMS FOR SEMILINEAR QUANTUM HARMONIC OSCILLATORS
Communications in Mathematical Physics, 2009Co-Authors: Benoît Grébert, Rafik Imekraz, Eric PaturelAbstract:We consider the semilinear harmonic oscillator $$i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d,\ t\in \R$$ where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least.\\ We prove that such a Hamiltonian Equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\geq 2$.\\ As a consequence we prove the almost global existence for solutions of the above Equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions.
Erwan Faou - One of the best experts on this subject based on the ideXlab platform.
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Hamiltonian Interpolation of Splitting Approximations for Nonlinear PDEs
Foundations of Computational Mathematics, 2011Co-Authors: Erwan Faou, Benoît GrébertAbstract:We consider a wide class of semilinear Hamiltonian partial differential Equations and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical trajectory remains at least uniformly integrable with respect to an eigenbasis of the linear operator (typically the Fourier basis). We show the existence of a modified interpolated Hamiltonian Equation whose exact solution coincides with the discrete flow at each time step over a long time. While for standard splitting or implicit–explicit schemes, this long time depends on a cut-off condition in the high frequencies (CFL condition), we show that it can be made exponentially large with respect to the step size for a class of modified splitting schemes.
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Hamiltonian interpolation of splitting approximations for nonlinear PDEs
Foundations of Computational Mathematics, 2011Co-Authors: Erwan Faou, Benoît GrébertAbstract:We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni- formly integrable with respect to an eigenbasis of the linear operator (typically the Fourier basis). We show the existence of a modified interpolated Hamiltonian Equation whose exact solution coincides with the discrete flow at each time step over a long time depending on a non resonance condition satisfied by the stepsize. We introduce a class of modified splitting schemes fulfilling this condition at a high order and prove for them that the numerical flow and the continuous flow remain close over exponentially long time with respect to the step size. For stan- dard splitting or implicit-explicit scheme, such a backward error analysis result holds true on a time depending on a cut-off condition in the high frequencies (CFL condition). This analysis is valid in the case where the linear operator has a discrete (bounded domain) or continuous (the whole space) spectrum.
Eric Paturel - One of the best experts on this subject based on the ideXlab platform.
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NORMAL FORMS FOR SEMILINEAR QUANTUM HARMONIC OSCILLATORS
Communications in Mathematical Physics, 2009Co-Authors: Benoît Grébert, Rafik Imekraz, Eric PaturelAbstract:We consider the semilinear harmonic oscillator $$i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d,\ t\in \R$$ where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least.\\ We prove that such a Hamiltonian Equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\geq 2$.\\ As a consequence we prove the almost global existence for solutions of the above Equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions.
Jeremy L. Marzuola - One of the best experts on this subject based on the ideXlab platform.
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Dispersive estimates using scattering theory for matrix Hamiltonian Equations
Discrete and Continuous Dynamical Systems, 2011Co-Authors: Jeremy L. MarzuolaAbstract:We develop the techniques of [25] and [11] in order to derive dispersive estimates for a matrix Hamiltonian Equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schrodinger Equation $\i u_t + \Delta u + \beta (|u|^2) u = 0$ $\u(0,x) = u_0 (x),$ in $\mathbb{R}^3$. These results have been seen before, though we present a new approach using scattering theory techniques. In further works, we will numerically and analytically study the existence of a minimal mass soliton, as well as the spectral assumptions made in the analysis presented here.
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Dispersive estimates using scattering theory for matrix Hamiltonian Equations
arXiv: Analysis of PDEs, 2009Co-Authors: Jeremy L. MarzuolaAbstract:We develop the techniques of \cite{KS1} and \cite{ES1} in order to derive dispersive estimates for a matrix Hamiltonian Equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schr\"odinger Equation {c} i u_t + \Delta u + \beta (|u|^2) u = 0 u(0,x) = u_0 (x), in $\reals^3$. These results have been seen before, though we present a new approach using scattering theory techniques. In further works, we will numerically and analytically study the existence of a minimal mass soliton, as well as the spectral assumptions made in the analysis presented here.
Sandrine Grellier - One of the best experts on this subject based on the ideXlab platform.
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Invariant tori for the cubic Szegö Equation
Inventiones Mathematicae, 2012Co-Authors: Patrick Gérard, Sandrine GrellierAbstract:We continue the study of the following Hamiltonian Equation on the Hardy space of the circle, $$i\partial _tu=\Pi(|u|^2u)\ ,$$ where $\Pi $ denotes the Szegö projector. This Equation can be seen as a toy model for totally non dispersive evolution Equations. In a previous work, we proved that this Equation admits a Lax pair, and that it is completely integrable. In this paper, we construct the action-angle variables, which reduces the explicit resolution of the Equation to a diagonalisation problem. As a consequence, we solve an inverse spectral problem for Hankel operators. Moreover, we establish the stability of the corresponding invariant tori. Furthermore, from the explicit formulae, we deduce the classification of orbitally stable and unstable traveling waves.
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The Szegö Cubic Equation
Annales Scientifiques de l'École Normale Supérieure, 2010Co-Authors: Patrick Gérard, Sandrine GrellierAbstract:We consider the following Hamiltonian Equation on the $L^2$ Hardy space on the circle, $$i\partial _tu=\Pi(|u|^2u)\ ,$$ where $\Pi $ is the Szegö projector. This Equation can be seen as a toy model for totally non dispersive evolution Equations. We display a Lax pair structure for this Equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating the degeneracy of this completely integrable structure. We also classify the traveling waves for this system.
