Introduction to Mathematical Physics/Some mathematical problems and their solution/Nonlinear evolution problems, perturbative methods

Problem statement
Perturbative methods allow to solve nonlinear evolution problems. They are used in hydrodynamics, plasma physics for solving nonlinear fluid models (see for instance . Problems of nonlinear ordinary differential equations can also be solved by perturbative methods (see for instance   where averaging method is presented). Famous KAM theorem (Kolmogorov--Arnold--Moser) gives important results about the perturbation of hamiltonian systems. Perturbative methods are only one of the possible methods: geometrical methods, normal form methods  can give good results. Numerical technics will be introduced at next section.

Consider the following problem:

Various perturbative methods are presented now.

Regular perturbation
Solving method can be described as follows:

This method is simple but singular problem my arise for which solution is not valid uniformly in $$t$$.

Born's iterative method
This method is more "global" than previous one \index{Born iterative method} and can thus suppress some divergencies. It is used in diffusion problems. It has the drawback to allow less control on approximations.

Multiple scales method
For examples see.

Poincaré-Lindstedt method
This method is closely related to previous one, but is specially dedicated to studying periodical solutions. Problem to solve should be:\index{Poincaré-Lindstedt}

Resolution method is the following:

WKB method
WKB (Wentzel-Krammers-Brillouin) method is also a perturbation method. It will be presented at section secWKB in the proof of ikonal equation.