What Does Weakly Nonlinear Mean in Wave Theory?

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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.
 
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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

\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>
<br /> <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&#039;t trust me on that, that weak and strong nonlinearities are more like qualifiers than rigorous definitions.
 
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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.
 
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.
 
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