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:

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

$u$ verifies boundary conditions on the border $\partial \Omega$ of $\Omega$ .
$u$ verifies initial conditions.

Various perturbative methods are presented now.

Regular perturbation

Solving method can be described as follows:

Algorithm:

Differential equation is written as:

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

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

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

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

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

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^[1] , 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:

Differential equation is transformed into an integral equation:

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

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:

Assume the system can be written as:

eqavece

${\frac {\partial u}{\partial t}}=Lu+\epsilon N(u)$
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 $n\in \{0,\dots ,N\}$ .
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}

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:

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

fix

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

↑ Indeed solution of equation:
${\ddot {y}}+y=\cos t$
is
$y(t)=Acost+B\sin t+{\frac {1}{2}}t\sin t$

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[1] Indeed solution of equation:
${\ddot {y}}+y=\cos t$
is
$y(t)=Acost+B\sin t+{\frac {1}{2}}t\sin t$