ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

16. Applications

1991 MS Classification: 58F39, 58F14, 76B15, 34A47, 76B25, Iooss, Gérard, Institut Universitaire de France, Valbonne, France Travelling water-waves, as a paradigm for bifurcations in reversible infinite dimensional “dynamical” systems

First we recall few properties of reversible systems, in finite or infinite dimensional spaces, of the form

\frac{dU}{dx} = F (U), SF (U) = −F (SU), S^{2} = 𝕀

near a symmetric equilibrium taken at 0. Considering bifurcation problems, i. e vector fields depending on a set of parameters, which are non hyperbolic near 0 for certain parameter values, we give necessary conditions under which center manifold reduction applies for the study of solutions staying close to the equilibrium for all x ∈ ℝ (see the basic paper [6] for application of center manifold reduction to elliptic problems in a strip). We give the basic example of ”reversible Hopf ” bifurcation and state the results on known small bounded solutions: periodic, quasi-periodic, homoclinic to 0, homoclinic to periodic solutions. The main tool for such a study is Poincaré-Birkhoff normal form theory [1] (revisited following the global characterization given in [2], taking care of reversibility) and a proof of persistence of reversible (symmetric) solutions under higher order terms, not considered in the normal form [5] (see [3], [7] for other typical proofs on a similar reversible bifurcation). We then present a large class of water-wave problems, where we look for 2D travelling waves in a potential flow, with one or several superposed layers, in the presence or not of surface tension at the free surface and at interfaces. Coordinate x corresponds to the direction of travelling waves. In case of finite depth layers, such a problem of finding small bounded solutions for all x ∈ ℝ, is shown to be reducible to a finite dimensional symmetric center manifold, on which the system reduces to a reversible ODE. For ”reversible Hopf ” bifurcation, this leads to various kind of travelling waves, for instance to solitary waves with damped oscillations at infinity or to ”black” solitary waves, corresponding to solutions homoclinic to periodic waves with a minimum of amplitude at x = 0. If the bottom layer has infinite depth, which appears to be the most physically realistic case, concerning the validity of results in the parameter set, the mathematical problem is far more difficult, because it cannot be reduced to a finite dimensional one, due to the occurence of a continuous spectrum of the linearized operator DF (0) crossing the imaginary axis (at 0). This continuous spectrum may also contain an embedded eigenvalue at 0. We give some ways, via an infinite dimensional normal form reduction, on how to attack this difficulty, specially for periodic and homoclinic solutions with a polynomial decay at infinity (see [4] for solitary waves in the case of one layer of infinite depth, in presence of surface tension). References

[1] V. Arnold. Geometrical Methods in the theory of ODE. Springer 1983.

[2] G. Iooss, M. Adelmeyer. Topics in bifurcation theory and applications. Adv. Series in Nonlinear Dynamics, 3, World Sci. 1992.

[3] G. Iooss, K. Kirchgässner. Water waves for small surface tension: an approach via normal form. Proc. Roy. Soc. Edinburgh, 122A, 267-299, 1992.

[4] G. Iooss, P. Kirrmann. Capillary gravity waves on the free surface of an invicid fluid of infinite depth. Existence of solitary waves. Arch. Rat. Mech. Anal. 136, 1-19, 1996.

[5] G. Iooss, M. C. Pérouème. Perturbed homoclinic solutions in reversaible 1:1 resonance vector fields. J. Diff. Equ. 102, 62-88, 1993.

[6] K. Kirchgässner. Wave solutions of reversible systems ans applications. J. Diff. Equ. 45, 113-127, 1982.

[7] E. Lombardi. Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Appplications to water waves. Arch. Rat. Mech. Anal. 137, 227-304, 1997.