Small amplitude limit

From DispersiveWiki

The small amplitude limit for a nonlinear equation arises when considering initial position u(0) of the form u(0) = εf for some fixed f and a small parameter ε > 0, in the limit . For equations which are second-order in time, such as nonlinear wave equations, one must also specify an initial velocity u_t(0) = εg.

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 / ε), significant nonlinear effects may still occur in the limit.