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

Problem statementEdit

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:


 

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

Various perturbative methods are presented now.

Regular perturbationEdit

Solving method can be described as follows:

Algorithm:


  1. Differential equation is written as:

 

  1. The solution   of the problem when   is zero is known.
  2. General solution is seeked 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  .

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 methodEdit

Algorithm:


  1. Differential equation is transformed into an integral equation:

 

  1. A sequence of functions   converging to the solution   is seeked:

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 methodEdit

Algorithm:


  1. Assume the system can be written as:

    eqavece

     

  2. Solution   is looked for as:

    eqdevmu

     

    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 methodEdit

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:


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

    fix

     

  2. Solutions are seeked as:

    form1

     

    form2

     

    with  .

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

WKB methodEdit

mathsecWKB

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:

     

    is