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

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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:

proeqp

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:

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 method[edit]

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 method[edit]

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 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}

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 method[edit]

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