Small amplitude limit

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The small amplitude limit for a nonlinear equation arises when considering initial position of the form for some fixed and a small parameter , in the limit . For equations which are second-order in time, such as nonlinear wave equations, one must also specify an initial velocity .

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