Erwan Faou

Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus

It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.

Hamiltonian Interpolation of Splitting Approximations for Nonlinear PDEs

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.

Landau Damping in Sobolev Spaces for the Vlasov-HMF Model

We consider the Vlasov-HMF (Hamiltonian Mean-Field) model. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that these solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping.

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I. Finite-dimensional discretization

We consider discretized Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite-dimensional Birkhoff normal form result, we show the almost preservation of the actions of the numerical solution associated with the splitting method over arbitrary long time and for asymptotically large level of space approximation, provided the Sobolev norm of the initial data is small enough. This result holds under generic non-resonance conditions on the frequencies of the linear operator and on the step size. We apply these results to nonlinear Schrödinger equations as well as the nonlinear wave equation.

Asymptotic preserving schemes for the Klein---Gordon equation in the non-relativistic limit regime

We consider the Klein–Gordon equation in the non-relativistic limit regime, i.e. the speed of light

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part II. Abstract splitting

c

Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation

c tending to infinity. We construct an asymptotic expansion for the solution with respect to the small parameter depending on the inverse of the square of the speed of light. As the first terms of this asymptotic can easily be simulated our approach allows us to construct numerical algorithms that are robust with respect to the large parameter