What Does Weakly Nonlinear Mean in Wave Theory?

[hanson]

Hi all.
I am reading things about wave theory.
I am rather confused about the term "weakly nonlinear".
Say for the KdV equation:
u_t + 6uu_x + u_xxx = 0
This shall be a nonlinear equation due to the term uu_x, right?
Is it a "weakly nonlinear" equation or what?
Is "weakly nonlinear" something related to the derivation of the KdV equation or that's something related to the way we solve this nonlinear equation?
I read a book which use a perturbation method to solve this equation, and it assume u to have a perturbtive expansion as follows:
u = eu1 + e^2u2 + e^3u3 + ...where e is the small perturbation.

Why don't it assumes
u = u0 + eu1 + e^2u2 + e^3u3 + ...?

Is there anything to do with "weakly nonlinearity"?

Please help.

 

[AiRAVATA]

In this article, I've found the following definition:

The initial value problem of the form

$$u''(t)+Au'(t)=F(t,u(t),u'(t))\qquad (1)$$

Throughout the paper is assumed that the nonlinear operator $F$ is Lipschitz continuous in $u$, and for this reason, (1) is said to be weakly nonlinear ($-A$ is a generator of a strongly continuous semigroup).

Also, according to the great book of Lawrence C. Evans Partial Differential equation, we have

Definitions

The partial differential equation

$$F(D^ku(x),D^{k-1}u(x),...,Du(x),u(x),x)=0 \qquad (2)$$

(i) Is called linear if it has the form

[tex]\sum_{|\alpha|\le k}a_\alpha(x)D^\alpha u=f(x)[/itex]<br /> <br /> <i>for given functions</i> $a_\alpha\,(\alpha\le k),\,f$. <i>This linear PDE is </i>homogeneous <i>if</i> $f \equiv 0$.<br /> <br /> (ii)<i> The PDE (2) is </i>semilinear<i> if it has the form</i><br /> <br /> $$\sum_{|\alpha|\le k}a_\alpha(x) D^\alpha u+a_0(D^{k-1}u,...,Du,u,x)=0.$$<br /> <br /> (iii) <i>The PDE (2) is </i>quasilinear<i> if it has the form</i><br /> <br /> $$\sum_{|\alpha\le k}a_\alpha(D^{k-1}u,...,Du,u,x)D^\alpha u +a_0(D^{k-1}u,...,Du,u,x)=0.$$<br /> <br /> (iv)<i> The PDE (2) is </i>fully nonlinear<i> if it depends nonlinearly upon the highest order derivatives.</i>[/tex]

$$<br /> Is my experience that this definitions are not well established among literature and people tend to refer to semilinearity and quasilinearity as nonlinearity. I also believe, but don't trust me on that, that weak and strong nonlinearities are more like qualifiers than rigorous definitions.$$

 

[hunt_mat]

Weakly nonlinear usually means that there is only one term which is nonlinear. Usually in the case of fluid flow (such as the KdV), the assumption is made of being a long wave length which when you do the asymptotic expansions, give rise to only one nonlinear term. That is what people generally refer to as weakly nonlinear.

 

[HallsofIvy]

By the way, the adjective phrase is "weakly nonlinear" but the noun phrase is "weak nonlinearity", not "weakly nonlinearity". "Weakly" is an adverb and cannot modify a noun.