This paper outlines the manner in which Thom’s theory of catastrophes fits into the Hamilton-Jacobi theory of partial differential equations. The representation of solutions of a first order partial differential equation as lagrangian manifolds allows one to study the local structure of their singularities. The structure of generic singularities is closely related to Thom’s concept of the elementary catastrophe associated to a singularity. Three concepts of the stability of a singularity are discussed....

We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps.

We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations    (SE) f(x,u,Dx u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(0,0,{\xi }^0)\in {ℂ}_x^n\times {ℂ}_u\times {ℂ}_{\xi }^n({\xi }^0=D_{x}u\left(0\right))$ and $f(0,0,{\xi }^0)=0$. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $\left(\xi \in {ℂ}^n\right)$. The criterion of convergence of a formal solution $u\left(x\right)={\sum }_{\left|\alpha \right|\ge 1}{u}_{\alpha }{x}^{\alpha }$ of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal...

In this paper we present a few results on convergence for the prime integrals equations connected with the bounce problem. This approach allows both to prove uniqueness for the one-dimensional bounce problem for almost all permissible Cauchy data (see also [6]) and to deepen previous results (see [3], [5], [7]).