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

## Problem statement

[edit | edit source]Perturbative methods allow to solve nonlinear evolution problems. They are used in hydrodynamics, plasma physics for solving nonlinear fluid models (see for instance ([ph:plasm:Chen84]). Problems of nonlinear ordinary differential equations can also be solved by perturbative methods (see for instance ([ma:equad:Arnold83]) 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 ([ma:equad:Arnold83]) can give good results. Numerical technics will be introduced at next section.

Consider the following problem:

proeqp

**Problem:**
Find such that:

- verifies boundary conditions on the border of .
- verifies initial conditions.

Various perturbative methods are presented now.

## Regular perturbation

[edit | edit source]Solving method can be described as follows:

**Algorithm:**

- Differential equation is written as:

- The solution of the problem when is zero is known.
- General solution is sought as:

- Function is developed around using Taylor type formula:

- A hierarchy of
*linear*equations to solve is obtained:

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

**Example:**

*Non uniformity of regular perturbative expansions* (see
([ma:equad:Bender87]).
Consider Duffing equation:

Let us look for solution which can be written as:

The linear hierarchy obtained with the previous assumption is:

With initial conditions:

,

one gets:

and a particular solution for will be unbounded^{[1]}
, now solution is expected to be bounded.
Indeed (see [ma:equad:Bender87]),
multiplying Duffing equation by
,
one gets the following differential equation:

We have thus:

where is a constant. Thus is bounded if .

**Remark:**

In fact Duffing system is conservative.

**Remark:**

*Origin of secular terms* : A regular perturbative expansion of a
periodical function whose period depends on a parameter gives rise
automatically to secular terms (see
([ma:equad:Bender87]):

## Born's iterative method

[edit | edit source]**Algorithm:**

- Differential equation is transformed into an integral equation:

- A sequence of functions converging to the solution is sought:

Starting from chosen solution , successive are evaluated using recurrence formula:

This method is more "global" than previous one \index{Born iterative method} and can thus suppress some divergencies. It is used in diffusion problems ([ph:mecaq:Cohen73],[ph:mecaq:Cohen88]). It has the drawback to allow less control on approximations.

## Multiple scales method

[edit | edit source]**Algorithm:**

For examples see ([ma:equad:Nayfeh95]).

## Poincaré-Lindstedt method

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

**Problem:**

Find such that:

eqarespo

where is a periodic function of pulsation . Setting ,

one gets:

Resolution method is the following:

**Algorithm:**

## WKB method

[edit | edit source]

mathsecWKB

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

- ↑
Indeed solution of equation:
is