Many problems in fluid mechanics involve asymptotic expansions in the form of a power series for a suitable small parameter. Such expansions necessarily fail to find terms which are exponentially small with respect to this parameter. Although small these missing terms are often of physical importance. This chapter will describe how to find such exponentially small terms, using as the main tool matched asymptotic expansions in the complex plane and Borel summation. The techniques are developed in the context of model problems related primarily to the theory of weakly nonlocal solitary waves (also called generalized solitary waves) which arise in the study of gravity-capillary waves, internal waves and in several other physical contexts. These waves have a central core of finite amplitude, but are accompanied by co-propagating oscillatory tails whose amplitude is exponentially small. Special interest lies in the possibility that for certain parameter values, the amplitude of the oscillatory tails may be zero, leading to the important concept of embedded solitary waves.
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