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

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Problem statement[edit]

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[edit]

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[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^2t^2}{2}+\dots)+(\epsilon t+\dots).cos(t)

Born's iterative method[edit]

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[edit]

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[edit]

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_0and substituting form1 and form2 into eqarespo.

WKB method[edit]

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)=A cos t+ B \sin t+\frac{1}{2}t \sin t