Small amplitude limit

The small amplitude limit for a nonlinear equation arises when considering initial position $u(0)$ of the form $u(0)=\epsilon f$ for some fixed $f$ and a small parameter $\epsilon >0$, in the limit $\epsilon \to 0$. For equations which are second-order in time, such as nonlinear wave equations, one must also specify an initial velocity $u_{t}(0)=\epsilon g$.

For bounded times, the small amplitude limit is usually just the linear counterpart of the equation; however when analyzing long times (e.g. times comparable to $1/\epsilon$), significant nonlinear effects may still occur in the limit.