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

Problem statement edit

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:


Problem: Find   such that:

  1.   verifies boundary conditions on the border   of  .
  2.   verifies initial conditions.

Various perturbative methods are presented now.

Regular perturbation edit

Solving method can be described as follows:


  1. Differential equation is written as:
  1. The solution   of the problem when   is zero is known.
  2. General solution is sought as:
  1. Function   is developed around   using Taylor type formula:
  1. 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  .


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  .


In fact Duffing system is conservative.


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


  1. Differential equation is transformed into an integral equation:
  1. 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


  1. Assume the system can be written as:


  2. Solution   is looked for as:



    with   for all  .

  3. A hierarchy of equations to solve is obtained by substituting expansion eqdevmu into equation eqavece.

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

Poincaré-Lindstedt method edit

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


Find   such that:



where   is a periodic function of pulsation  . Setting  ,

one gets:


Resolution method is the following:


  1. Existence of a solution   which does not depend on   is imposed:


  2. Solutions are sought as:





    with  .

  3. A hierarchy of linear equations to solve is obtained by expending   around  and substituting form1 and form2 into eqarespo.

WKB method edit


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

  1. Indeed solution of equation: