Title: Large Phenomena beyond all orders and bifurcation of reversible
homoclinic connections. Applications to water waves.

Speaker: Eric Lombardi, INLN, Nice

In many physical problems one can construct ``solutions''  in term of power
series of a small parameter $\epsilon$ which reads
$ Y(t,\epsilon) = \sum\limits_{n= 0}^k \epsilon^n Y_n(t) + o(\epsilon^k).$
In some cases, this power series  is defined at any order but  it diverges.
This divergence may express that the system has a solution for which such an
expansion misses some exponentially small term  like $e^{-1/\epsilon^2}
Z(t)$.   {\em Such a term  is said to lie  beyond any algebraic
orders}.  This kind of phenomena typically occurs for physical problems
governed by reversible vector fields  near resonances because of the
degeneracy induced by the symmetry and the resonance. Such  problems, 
in which these very small terms have great practical interest, are
known  in many branches of science including dendritic crystals growth,
quantum tunneling, KAM theory, theory of water-waves (which was our
original motivation for this work) and others. A careful study of these
problem reveals that in most of the cases, the exponentially small terms
come from {\em oscillatory integrals}. To study such problems, we have  
developed a general method to obtain an exponentially small equivalent 
of  oscillatory integrals involving solutions of  {\em nonlinear differential
equations}. The method is based on a very precise description of the 
holomorphic continuation of the solutions of the perturbed equation and 
in particular of their behavior  near the complex singularities 
of the solutions of the unperturbed system. The method that we use to 
build step by step  the holomorphic continuation of the solutions is a 
rigorous version for this problem of the formal theory of Matching 
Asymptotic Expansions (M.A.E.). In particular, the construction of the 
holomorphic continuation of the solutions near the singularities is based 
on the study of an ``inner system'' which is well known in the theory of 
M.A.E. and which appears to be the relevant part of the equation near 
the singularities. 

This method enables us to study the problem of the persistence of 
homoclinic connections to 0 for the one parameter families $V(.,\mu)$
of reversible vector  fields which admit at the origin a $0^2i\omega$ 
resonance, i.e. vector fields for which the origin is a fixed point 
and for which $0$ is a double non semi simple eigenvalues and $\pm i\omega$ 
are simple eigenvalues of the differential at the origin $DV_u(0,0)$.
This problem cannot be solved by a direct Melnikov approach since 
the Melnikov function is given by an oscillatory integral and is 
exponentially small.  Our ``Exponential tools'' enable us to prove [2] 
that  generically, vector fields admitting at 0 a $0^{2}i\omega$ resonance 
at the origin  do not admit any  homoclinic connection to 0, whereas we  
proved in [1] that they always admit homoclinic connections to exponentially 
small periodic orbits. One example of such a vector field  in infinite 
dimensions occurs when describing the irrotational flow of an inviscid 
fluid layer under the influence of gravity and small surface tension (Bond 
number $b<1/3$) for a Froude number $F$ close to $1$. In this context a 
homoclinic connection to 0  would be called a  solitary wave. We have also 
developed a general method to obtain an exponentially small equivalent of  
bi-oscillatory integrals. In this case, the interaction between the two 
frequencies, makes the  study more intricate and a partial complexification 
of time is necessary. This last method enables to study the $(i\omega_0)^2 
i\omega_1$ resonance, i.e. one parameter families $V(\cdot, \mu)$ of 
reversible vector fields for which $\pm i \omega_0$ are a double non 
semi simple eigenvalues and $\pm i\omega_1$ are simple eigenvalues of the 
differential at the origin $DV_u(0,0)$. Such a resonance occurs  for  
water waves when studying  several layers problems  and for chains of 
coupled nonlinear oscillators.

[1] Lombardi E. Homoclinic orbits to exponentially small periodic orbits 
for a class of reversible  systems.   Application to   water waves. 
{\em Arch. Rational Mech.  Anal.},  {\bf 137}:p. 227-304, (1997).
[2] Lombardi E.  Oscillatory integrals and  phenomena beyond any
    algebraic order; with applications to homoclinic orbits in reversible 
    systems.  Book in preparation.