# What is weakly nonlinearity?

1. Apr 10, 2007

### hanson

Hi all.
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"?

2. Apr 10, 2007

### AiRAVATA

The initial value problem of the form

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

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

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.

Last edited: Apr 10, 2007
3. Jun 26, 2010

### 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.

4. Jun 27, 2010

### 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.