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138 3. Further applications

where Again we expand

for each Q; immediately we see that the leading-order terms satisfy

(cf. (3.71)), but these functions are otherwise unknown.
The terms are defined by the equations

from (3.72)–(3.75), respectively, and this last equation is used only to define The
first three equations involve the combinations of terms:

respectively, and so we may eliminate all of and between them, which we
do. The resulting single equation involving and is written in terms
of one function— say—by using (3.77); this gives the leading term (for U) in the
far-field as the solution of

Thus is described by an equation in which the time of the
solution is controlled by both nonlinearity and dissipation cf. the