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

## 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

Problem: Find ${\displaystyle u\in V}$  such that:

${\displaystyle {\frac {\partial u}{\partial t}}=Lu+N(u),u\in E,x\in \Omega }$

1. ${\displaystyle u}$  verifies boundary conditions on the border ${\displaystyle \partial \Omega }$  of ${\displaystyle \Omega }$ .
2. ${\displaystyle u}$  verifies initial conditions.

Various perturbative methods are presented now.

## Regular perturbation

Solving method can be described as follows:

Algorithm:

1. Differential equation is written as:

${\displaystyle {\frac {\partial u}{\partial t}}=Lu+\epsilon N(u)}$

1. The solution ${\displaystyle u_{0}}$  of the problem when ${\displaystyle \epsilon }$  is zero is known.
2. General solution is seeked as:

${\displaystyle u(t)=\sum \epsilon ^{i}u_{i}(t)}$

1. Function ${\displaystyle N(u)}$  is developed around ${\displaystyle u_{0}}$  using Taylor type formula:

${\displaystyle N(u_{0}+\epsilon u_{1})=N(u_{0})+\epsilon u_{1}\left.{\frac {\partial N}{\partial u}}\right)_{u_{0}}}$

1. A hierarchy of linear equations to solve is obtained:

${\displaystyle {\frac {\partial u_{0}}{\partial t}}=Lu_{0}}$

${\displaystyle {\frac {\partial u_{1}}{\partial t}}=Lu_{1}+u_{1}\left.{\frac {\partial N}{\partial u}}\right)_{u_{0}}}$

This method is simple but singular problem my arise for which solution is not valid uniformly in ${\displaystyle t}$ .

Example:

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

${\displaystyle {\frac {d^{2}y}{dt^{2}}}+y+\epsilon y^{3}=0}$

Let us look for solution ${\displaystyle y(t)}$  which can be written as:

${\displaystyle y(t)=y_{0}(t)+\epsilon y_{1}(t)+\epsilon ^{2}y_{2}(t)}$

The linear hierarchy obtained with the previous assumption is:

${\displaystyle {\begin{matrix}{\frac {d^{2}y_{0}}{dt^{2}}}+y_{0}&=&0\\{\frac {d^{2}y_{1}}{dt^{2}}}+y_{1}&=&-y_{0}^{3}\end{matrix}}}$

With initial conditions:

${\displaystyle y_{0}(0)=1,y'_{0}(0)=0}$ ,

one gets:

${\displaystyle y_{0}(t)=cos(t)}$

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

${\displaystyle {\frac {d}{dt}}[{\frac {1}{2}}({\frac {dy}{dt}})^{2}+{\frac {1}{2}}y^{2}+{\frac {1}{4}}\epsilon y^{4}]=0.}$

We have thus:

${\displaystyle {\frac {1}{2}}({\frac {dy}{dt}})^{2}+{\frac {1}{2}}y^{2}+{\frac {1}{4}}\epsilon y^{4}=C}$

where ${\displaystyle C}$  is a constant. Thus ${\displaystyle y^{2}}$  is bounded if ${\displaystyle \epsilon >0}$ .

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]):

${\displaystyle \sin((1+\epsilon )t)=\sin(t)cos(\epsilon t)+\sin(\epsilon ).cos(t)}$

${\displaystyle =\sin(t)(1+{\frac {\epsilon ^{2}t^{2}}{2}}+\dots )+(\epsilon t+\dots ).cos(t)}$

## Born's iterative method

Algorithm:

1. Differential equation is transformed into an integral equation:

${\displaystyle u=\int _{0}^{t}(Lu+N(u))dt'}$

1. A sequence of functions ${\displaystyle u_{n}}$  converging to the solution ${\displaystyle u}$  is seeked:

Starting from chosen solution ${\displaystyle u_{0}}$ , successive ${\displaystyle u_{n}}$  are evaluated using recurrence formula:

${\displaystyle u_{n+1}=\int _{0}^{t}(Lu_{n}+N(u_{n}))dt'}$

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

Algorithm:

1. Assume the system can be written as:

eqavece

${\displaystyle {\frac {\partial u}{\partial t}}=Lu+\epsilon N(u)}$

2. Solution ${\displaystyle u}$  is looked for as:

eqdevmu

${\displaystyle u(x,t)=u_{0}(x,T_{0},T_{1},\dots ,T_{N})+\epsilon u(x,T_{0},T_{1},\dots ,T_{N})+\dots +O(\epsilon ^{N})}$

with ${\displaystyle T_{n}=\epsilon ^{n}t}$  for all ${\displaystyle n\in \{0,\dots ,N\}}$ .

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

## 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}

Problem:

Find ${\displaystyle u}$  such that:

eqarespo

${\displaystyle G(u,\omega )=0}$

where ${\displaystyle u}$  is a periodic function of pulsation ${\displaystyle \omega }$ . Setting ${\displaystyle \tau =\omega t}$ ,

one gets:

${\displaystyle u(x,\tau +2\pi )=u(x,\tau )}$

Resolution method is the following:

Algorithm:

1. Existence of a solution ${\displaystyle u_{0}(x)}$  which does not depend on ${\displaystyle \tau }$  is imposed:

fix

${\displaystyle G(u_{0}(x),\omega )=0}$

2. Solutions are seeked as:

form1

${\displaystyle u(x,\tau ,\epsilon )=u_{0}(x)+\epsilon u_{1}(x,\tau )+{\frac {\epsilon ^{2}}{2}}u_{2}(x,\tau )+\dots }$

form2

${\displaystyle \omega (\epsilon )=\omega _{0}+\epsilon \omega _{1}+{\frac {\epsilon }{2}}\omega _{2}}$

with ${\displaystyle u(x,\tau ,\epsilon =0)=u_{0}(x)}$ .

3. A hierarchy of linear equations to solve is obtained by expending ${\displaystyle G}$  around ${\displaystyle u_{0}}$ 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.

1. Indeed solution of equation:

${\displaystyle {\ddot {y}}+y=\cos t}$

is

${\displaystyle y(t)=Acost+B\sin t+{\frac {1}{2}}t\sin t}$