The major target is to provide on the point of novices a number of glossy instruments of micro-local research that are necessary for the mathematical examine of nonlinear partial differential equations. The center of those notes is dedicated to a presentation of the para-differential recommendations, which mix a linearization process for nonlinear equations, and a symbolic calculus which mimics or extends the classical calculus of Fourier multipliers. those tools practice to many difficulties in nonlinear PDE’s resembling elliptic equations, propagation of singularities, boundary price difficulties, shocks or boundary layers. despite the fact that, in those introductory notes, we've selected to demonstrate the speculation on chosen and comparatively uncomplicated examples, which permit changing into acquainted with the concepts. They trouble the good posed-ness of the Cauchy challenge for structures of nonlinear PDE's, to start with hyperbolic platforms and secondly coupled structures of Schrödinger equations which come up in numerous versions of wave propagation.
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