What is weakly nonlinearity?

[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]

for given functions $a_\alpha\,(\alpha\le k),\,f$. This linear PDE is homogeneous if $f \equiv 0$.

(ii) The PDE (2) is semilinear if it has the form

$$\sum_{|\alpha|\le k}a_\alpha(x) D^\alpha u+a_0(D^{k-1}u,...,Du,u,x)=0.$$

(iii) The PDE (2) is quasilinear if it has the form

$$\sum_{|\alpha\le k}a_\alpha(D^{k-1}u,...,Du,u,x)D^\alpha u +a_0(D^{k-1}u,...,Du,u,x)=0.$$

(iv) The PDE (2) is fully nonlinear if it depends nonlinearly upon the highest order derivatives.

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.

 

[blue_raver22]

I can provide you with a response to your question about weakly nonlinearity. In simple terms, weakly nonlinearity refers to a system or equation where the nonlinear effects are small compared to the linear effects. In the case of the KdV equation, the nonlinear term uu_x is small compared to the linear term u_xxx, making it a weakly nonlinear equation.

The use of the perturbation method in solving this equation is related to the fact that the nonlinear effects are small, so we can approximate the solution by expanding it in a series of small perturbations. This is why the book you read assumes u to have a perturbative expansion, as it is a valid approach for solving weakly nonlinear equations.

As for your question about why the perturbation expansion does not start with u0, it is because the linear term u_xxx does not have a coefficient of e. In a perturbation expansion, we assume that the solution can be written as a sum of terms, each multiplied by a power of e. Since the linear term does not have a coefficient of e, it is not included in the perturbation expansion.

In conclusion, weakly nonlinearity is a concept that is related to the size of nonlinear effects in a system or equation. It is an important consideration in many scientific fields, including wave theory. I hope this helps clarify your confusion.