A PARADIGM FOR THE CREATION OF SCALES AND PHASES IN NONLINEAR EVOLUTION EQUATIONS
Résumé
The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly in a nonlinear flow under the impact of forcing terms. This phenomenon is illustrated in the context of Hamilton-Jacobi equations. It is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. We introduce an original method allowing to get the existence of solutions during
long times, as well as asymptotic descriptions and reduced models. This is achieved by implementing three main tools: a blow-up procedure (extending the normal form method), a Wentzel-Kramers-Brillouin approximation (of super-critical type) and the Hadamard's global inverse function theorem which can be applied here due to transparency conditions (coming from a complete integrability condition).
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