What Does Weakly Nonlinear Mean in Wave Theory?

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The term "weakly nonlinear" in wave theory refers to systems where nonlinearity is present but not dominant, allowing for perturbative methods to be applied. In the context of the KdV equation, the presence of the term uu_x indicates nonlinearity, but the weakly nonlinear classification suggests that the effects of nonlinearity can be treated as small perturbations. The perturbation method used in solving the KdV equation assumes a small parameter e to expand the solution, indicating that the nonlinearity is not strong enough to complicate the analysis significantly. The choice of expansion starting from u = eu1 rather than u = u0 + eu1 is linked to the assumption that the leading order term is negligible in the weakly nonlinear regime. Understanding these concepts is crucial for applying appropriate mathematical techniques to analyze wave behavior in weakly nonlinear systems.
hanson
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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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