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:

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

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:

$\begin{matrix} \frac{d^2y_0}{dt^2}+y_0&=&0\\ \frac{d^2y_1}{dt^2}+y_1&=&-y_0^3 \end{matrix}$

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]):

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

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

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.

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:

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.