# Small amplitude limit

The small amplitude limit for a nonlinear equation arises when considering initial position ${\displaystyle u(0)}$ of the form ${\displaystyle u(0)=\epsilon f}$ for some fixed ${\displaystyle f}$ and a small parameter ${\displaystyle \epsilon >0}$, in the limit ${\displaystyle \epsilon \to 0}$. For equations which are second-order in time, such as nonlinear wave equations, one must also specify an initial velocity ${\displaystyle u_{t}(0)=\epsilon g}$.
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 ${\displaystyle 1/\epsilon }$), significant nonlinear effects may still occur in the limit.