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.

 

[Aero Stud]

Math forum maybe ?

 

[ravenprp]

Weakly nonlinearity refers to a situation where the nonlinearity in an equation is not strong enough to significantly affect the overall behavior of the system. In other words, the nonlinear terms are small compared to the linear terms in the equation. This can be seen in the KdV equation you mentioned, where the term uu_x is relatively small compared to the other terms.

In terms of solving the equation, a perturbation method is often used in weakly nonlinear systems. This involves assuming a small perturbation in the solution, as shown in the expansion u = eu1 + e^2u2 + e^3u3 + ..., where e is a small parameter. This allows for a simpler and more manageable solution compared to considering the full nonlinear equation.

As for why the book you read only assumed a perturbation in the higher order terms (e^2, e^3, etc.), it could be that the system is only weakly nonlinear for small perturbations and therefore the higher order terms can be neglected. This is not always the case and it ultimately depends on the specific system being studied.

In summary, "weakly nonlinearity" refers to a situation where nonlinear terms are small compared to linear terms, and it can affect the way we solve nonlinear equations using perturbation methods.