# 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 $u\in V$  such that:

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

1. $u$  verifies boundary conditions on the border $\partial \Omega$  of $\Omega$ .
2. $u$  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:

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

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

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

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

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

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

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

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

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

Example:

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

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

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

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

The linear hierarchy obtained with the previous assumption is:

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

With initial conditions:

$y_{0}(0)=1,y'_{0}(0)=0$ ,

one gets:

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

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

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

We have thus:

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

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

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]):

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

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

## Born's iterative method

Algorithm:

1. Differential equation is transformed into an integral equation:

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

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

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

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

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

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

2. Solution $u$  is looked for as:

eqdevmu

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

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

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 $u$  such that:

eqarespo

$G(u,\omega )=0$

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

one gets:

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

Resolution method is the following:

Algorithm:

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

fix

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

2. Solutions are seeked as:

form1

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

form2

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

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

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

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

is

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