Small divisor problem in the theory of three-dimensional water gravity waves

We consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity $g$ and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle $2\theta$ between them. \newline Denoting by $\mu =gL/c^{2}$ the dimensionless bifurcation parameter ($L$ is the wave length along the direction of the travelling wave and $c$ is the velocity of the wave), bifurcation occurs for $\mu =\cos \theta$. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. "Diamond waves" are a particular case of such waves, when they are symmetric with respect to the direction of propagation.\newline \emph{The main object of the paper is the proof of existence} of such symmetric waves having the above mentioned asymptotic expansion. Due to the \emph{occurence of small divisors}, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles $\theta$, the 3-dimensional travelling waves bifurcate for a set of "good" values of the bifurcation parameter having asymptotically a full measure near the bifurcation curve in the parameter plane $(\theta ,\mu ).$Comment: 119

Small divisor problems in Fluid Mechanics. In memory of Klaus Kirchgässner

International audienceSeveral small divisor problems occuring in Fluid Mechanics are presented. Two of them come from water waves: 3D periodic travelling gravity waves, and 2D standing gravity waves. The last example comes from quasipatterns observed for thin viscous horizontal fluid layers periodically vertically shaked (Faraday type experiment)

Normal forms with exponentially small remainder : application to homoclinic connections for the reversible 0 2+ iω resonance

International audienceIn this note we explain how the normal form theorem established in [2] for analytic vector fields with a semi-simple linearization enables to prove the existence of homoclinic connections to exponentially small periodic orbits for reversible analytic vector fields admitting a 0 2+ iω resonance where the linearization is precisely not semi simple

Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems

International audienceThis book is an extension of different lectures given by the authors during manyyears at the University of Nice, at the University of Stuttgart in 1990, and the Uni-versity of Bordeaux in 2000 and 2001. Large parts of the first four chapters are ofmaster level and contain various examples and exercises, partly posed at exams.However, the infinite-dimensional set-up in Chapter 2 requires several tools andresults from the theory of linear operators. A brief description of these tools andresults is given in Appendix

Quasipatterns in a parametrically forced horizontal fluid film

International audienceWe shake harmonically a thin horizontal viscous fluid layer (frequency forcing Ω, only one harmonic), to reproduce the Faraday experiment and using the system derived in [31] invariant under horizontal rotations. When the physical parameters are suitably chosen, there is a critical value of the amplitude of the forcing such that instability occurs with at the same time the mode oscillating at frequency Ω/2, and the mode with frequency Ω. Moreover, at criticality the corresponding wave lengths kc and k′c are such that if we define the family of 2q equally spaced (horizontal) wave vectors kj on the circle of radius kc , then kj + kl = k′n, with |kj| = |kl| = kc , |k′n| = k′c .It results under the above conditions that 0 is an eigenvalue of the linearized operator in a space of time-periodic functions (frequencyΩ/2) having a spatially quasiperiodic pattern if q ≥ 4. Restricting our study to solutions invariant under rotations of angle 2π/q, gives a kernel of dimension 4.In the spirit of Rucklidge and Silber (2009) [29] we derive formally amplitude equations for perturbations possessing this symmetry. Then we give simple necessary conditions on coefficients, for obtaining the bifurcation of (formally) stable time-periodic (frequency Ω/2) quasipatterns. In particular,we obtain a solution such that a time shift by half the period, is equivalent to a rotation of angle π/q of the pattern

Asymmetrical three-dimensional travelling gravity waves

International audienceWe consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid. The pattern is non symmetric with respect to the propagation direction of the waves and we consider a general non resonant situation. Defining a couple of amplitudes ε 1 , ε 2 along the basis of wave vectors which satisfy the dispersion relation, first we give the formal asymptotic expansion of bifurcating solutions in powers of ε 1 , ε 2. Then, introducing an additional equation for the unknown diffeomorphism of the torus, associated with an irrational rotation number, which allows to transform the differential at the successive points of the Newton iteration method, into a differential equation with two constant main coefficients, we are able to use a descent method leading to an invertible differential. Then by using an adapted Nash Moser theorem, we prove the existence of solutions with the above asymptotic expansion, for values of the couple (ε1^2 , ε2^2) in a subset of the first quadrant of the plane, with asymptotic full measure at the origin

Multimodal Standing Gravity Waves: a Completely Resonant System

International audienceThe standing gravity wave problem on an infinitely deep fluid layer is considered under the form of a nonlinear non local scalar PDE of second order as in [6]. Nonreso-nance at quadratic order of the infinite dimensional bifurcation equation, allows to give the explicit form of the quadratic change of variables able to suppress quadratic terms in the nonlinear equation. We state precisely the equivalence between formulations in showing that the above unbounded change of variable is invertible. The infinite set of solutions which can be expanded in powers of amplitude ε is then given up to order ε 2

J. Boussinesq and the standing water waves problem

International audienceIn this short note we present the original Boussinesq's contribution to the nonlinear theory of the two dimensional standing gravity water wave problem, which he defined as " le clapotis "

Polynomial normal forms with exponentially small remainder for analytic vector fields

International audienceA key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms, invented by Poincaré, which gives simple forms to which a vector field can be reduced close to the equilibrium. In the class of formal vector valued vector fields the problem can be easily solved, whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs. Nevertheless the study of the dynamics in a neighborhood of the origin, can very often be carried out via a normalization up to finite order. This paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in R m. More precisely we prove that for any vector field in R m admitting the origin as a fixed point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small.We also give several examples of non semi-simple linearization for which this result is still true