Mathieu Colin

Stability and instability results for standing waves of quasi-linear Schrödinger equations

We study a class of quasi-linear Schrodinger equations arising in the theory of superfluid film in plasma physics. Using gauge transforms and a derivation process we solve, under some regularity assumptions, the Cauchy problem. Then, by means of variational methods, we study the existence, the orbital stability and instability of standing waves which minimize some associated energy.

ON THE LOCAL WELL-POSEDNESS OF QUASILINEAR SCHRÖDINGER EQUATIONS IN ARBITRARY SPACE DIMENSION

ABSTRACT We study the Cauchy Problem associated to a class of quasilinear Schrödinger equations which have been derived as models of several physical phenomenas. We prove local existence in arbitrary space dimension N without any smallness condition on the initial data.

Bifurcation from Semitrivial Standing Waves and Ground States for a System of Nonlinear Schrödinger Equations

We consider a system of nonlinear Schrodinger equations related to the Raman amplification in a plasma. We study the orbital stability and instability of standing waves bifurcating from the semitrivial standing wave of the system. The stability and instability of the semitrivial standing wave at the bifurcation point are also studied. Moreover, we determine the set of the ground states completely.

Solitons in quadratic media

In this paper, we investigate the properties of solitonic structures arising in quadratic media. First, we recall the derivation of systems governing the interaction process for waves propagating in such media and we check the local and global well-posedness of the corresponding Cauchy problem. Then, we look for stationary states in the context of normal or anomalous dispersion regimes, that lead us to either elliptic or non-elliptic systems and we address the problem of orbital stability. Finally, some numerical experiments are carried out in order to compute localized states for several regimes and to study dynamic stability as well as long-time asymptotics.

Numerical simulations of wormlike micelles flows in micro-fluidic T-shaped junctions

Numerical simulations of non-Newtonian fluids such as wormlike micellar solutions in confined geometries are of great interest in the oil industry. Their main property called shear-banding is a brutal transition from a very viscous state to a very fluid state above a certain threshold value of shear stress. This feature leads to a very complex behavior in 3D flows.A modified version of the Johnson-Segalmans model, adapted to our situation (flows with a strong extensional component) is presented. A particular attention is paid to inlet and outlet boundary conditions, and a Poiseuille-like submodel is derived in order to get natural velocity and stress profiles that can be used at the boundaries. A numerical method is then developed, and stability issues are presented.Our results show the interest of the modified Johnson-Segalmans model performed in this article. A set of 3D numerical simulations are then presented in order to understand the influence of the junction geometry upon the jamming effects observed with this kind of fluids.

Discrete asymptotic equations for long wave propagation

In this paper, we present a new systematic method to obtain some discrete numerical models for incompressible free-surface flows. The method consists in first discretizing the Euler equations with respect to one variable, keeping the other ones unchanged and then performing an asymptotic analysis on the resulting system. For the sake of simplicity, we choose to illustrate this method in the context of the Peregrine asymptotic regime, that is we propose an alternative numerical scheme for the so-called Peregrine equations. We then study the linear dispersion characteristics of our new scheme and present several numerical experiments to measure the relevance of the method.