From Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck Systems to Incompressible Euler Equations: the case with finite charge

We study the asymptotic regime of strong electric fields that leads from the Vlasov-Poisson system to the Incompressible Euler equations. We also deal with the Vlasov-Poisson-Fokker-Planck system which induces dissipative effects. The originality consists in considering a situation with a finite total charge confined by a strong external field. In turn, the limiting equation is set in a bounded domain, the shape of which is determined by the external confining potential. The analysis extends to the situation where the limiting density is non-homogeneous and where the Euler equation is replaced by the Lake Equation, also called Anelastic Equation.Comment: 39 pages, 3 figure

The KdV/KP-I Limit of the Nonlinear Schrödinger Equation

International audienceWe justify rigorously the convergence of the amplitude of solutions of nonlinear Schrödinger-type equations with nonzero limit at infinity to an asymptotic regime governed by the Korteweg-de Vries (KdV) equation in dimension 1 and the Kadomtsev-Petviashvili I (KP-I) equation in dimensions 2 and greater. We get two types of results. In the one-dimensional case, we prove directly by energy bounds that there is no vortex formation for the global solution of the nonlinear Schrödinger equation in the energy space and deduce from this the convergence toward the unique solution in the energy space of the KdV equation. In arbitrary dimensions, we use a hydrodynamic reformulation of the nonlinear Schrödinger equation and recast the problem as a singular limit for a hyperbolic system. We thus prove that smooth H^s solutions exist on a time interval independent of the small parameter. We then pass to the limit by a compactness argument and obtain the KdV/KP-I equation

Rarefaction pulses for the Nonlinear Schrödinger Equation in the transonic limit.

International audienceWe investigate the properties of finite energy travelling waves to the nonlinear Schrödinger equation with nonzero conditions at infinity for a wide class of nonlinearities. In space dimension two and three we prove that travelling waves converge in the transonic limit (up to rescaling) to ground states of the Kadomtsev-Petviashvili equation. Our results generalize an earlier result of F. Béthuel, P. Gravejat and J-C. Saut for the two-dimensional Gross-Pitaevskii equation, and provide a rigorous proof to a conjecture by C. Jones and P. H. Roberts about the existence of an upper branch of travelling waves in dimension three

Smooth branch of travelling waves for the Gross-Pitaevskii equation in R2\mathbb{R}^2 for small speed

We construct a smooth branch of travelling wave solutions for the 2 dimensional Gross-Pitaevskii equations for small speed. These travelling waves exhibit two vortices far away from each other. We also compute the leading order term of the derivatives with respect to the speed. We construct these solutions by an implicit function type argument

Travelling waves for the Nonlinear Schrödinger Equation with nonzero condition at infinity.

International audienceWe present two constraint minimization approaches to prove the existence of traveling waves for a wide class of nonlinear Schrödinger equations with nonvanishing conditions at infinity in space dimension N ≥ 2. Minimization of the energy at fixed momentum can be used whenever the associated potential function is positive on the natural function space and it gives a set of orbitally stable traveling waves. Minimization of the action at constant kinetic energy can be used in all cases, but gives no information on the orbital stability of the set of solutions

Coercivity for travelling waves in the Gross-Pitaevskii equation in R2\mathbb{R}^2 for small speed

In the previous paper, we constructed a smooth branch of travelling waves for the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space. In particular, our result proves the non degeneracy of these travelling waves, which is a key step in the classification of these waves and for the construction of multi-travelling waves

Long wave asymptotics for the Euler–Korteweg system

International audienceThe Euler–Korteweg system (EK) is a fairly general nonlinear waves model in mathematical physics that includes in particular the fluid formulation of the NonLinear Schrödinger equation (NLS). Several asymptotic regimes can be considered, regarding the length and the amplitude of waves. The first one is the free wave regime, which yields long acoustic waves of small amplitude. The other regimes describe a single wave or two counter propagating waves emerging from the wave regime. It is shown that in one space dimension those waves are governed either by inviscid Burgers or by Korteweg-de Vries equations, depending on the spatio-temporal and amplitude scalings. In higher dimensions, those waves are found to solve Kadomtsev-Petviashvili equations. Error bounds are provided in all cases. These results extend earlier work on defocussing (NLS) (and more specifically the Gross–Pitaevskii equation), and sheds light on the qualitative behavior of solutions to (EK), which is a highly nonlinear system of PDEs that is much less understood in general than (NLS)

Stationary solutions with vacuum for a one-dimensional chemotaxis model with non-linear pressure

International audienceIn this article, we study a one-dimensional hyperbolic quasi-linear model of chemotaxis with a non-linear pressure and we consider its stationary solutions, in particular with vacuum regions. We study both cases of the system set on the whole line \Er and on a bounded interval with no-flux boundary conditions. In the case of the whole line \Er, we find only one stationary solution, up to a translation, formed by a positive density region (called bump) surrounded by two regions of vacuum. However, in the case of a bounded interval, an infinite of stationary solutions exists, where the number of bumps is limited by the length of the interval. We are able to compare the value of an energy of the system for these stationary solutions. Finally, we study the stability of these stationary solutions through numerical simulations

Reconstruction of hydrometric data using a network optimization model

Water resource managers need to implement precise and efficient water management methods particularly in the context of low flow water management. The management objectives are complex because managers must satisfy both water demands for human activities and environmental goals. More often the flow objectives are defined at specific strategic points in which hydrometric stations are based. In order to allow the manager to better understand the hydrographical network behavior, in particular for inter-basin water transfer, these strategic hydrometric stations must be reinforced by some intermediate hydrometric stations, by modeling the network behavior, and by introducing weather forecast data in order to simulate the evolution in time and space of the river. For an efficient management, it is essential that the data collected and the output of the models (the natural flow and the withdrawals) must be reliable. For this purpose, a network optimization model was developed for analyzing the consistency of the available data set (measurements and model outputs) on a hydraulic system. Herein, a reconstruction of hydrometric data using this network optimization model is applied to the Arrats watershed management