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where we have again used subscripts to denote partial derivatives, and here the speed
(associated with the left side of the equation) is one. (This equation is usually called the
Boussinesq equation and it happens to be one of the equations that is completely integrable,
in the sense of soliton theory, for all see johnson, 1997, and Drazin & Johnson,
1992.) Our intention is to find an asymptotic solution of (3.9), valid as subject
to the initial data

where as (and, further, all relevant derivatives of f(x) satisfy this
same requirement).
The equation for (where u is the amplitude of the wave), on using our
familiar ‘iteration’ argument, suggests that we may seek a solution in the form

for some xs (distance) and some ts (time). Thus we obtain from (3.9)

(where ‘= 0’ means zero to all orders in which gives

and so on. The initial data, (3.10), then requires

and

Equation (3.12a) is the classical wave equation with the general solution (d’Alembert’s
solution)

for arbitrary functions F and G; application of the initial data for (given in (3.13))
then produces

(We note, in this example, that the particular initial data which we have been given
produces a wave moving only to the right (with speed 1), at this order.)