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

__toc__ 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 (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 Find such that: verifies boundary conditions on the border of . verifies initial conditions. Various perturbative methods are presented now. Regular perturbation Solving method can be described as follows: 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 . 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 unboundedIndeed solution of equation: is , 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 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 <li>Assume the system can be written as: eqavece <li>Solution is looked for as: eqdevmu with for all . <li>A hierarchy of equations to solve is obtained by substituting expansion eqdevmu into equation eqavece. For examples see (ma:equad:Nayfeh95). 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} Find such that: eqarespo where is a periodic function of pulsation . Setting , one gets: Resolution method is the following: Existence of a solution which does not depend on is imposed: fix Solutions are sought as: form1 form2 with . A hierarchy of linear equations to solve is obtained by expending around and substituting form1 and form2 into eqarespo. WKB method mathsecWKB WKB (Wentzel-Krammers-Brillouin) method is also a perturbation method. It will be presented at section secWKB in the proof of ikonal equation.