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follow the right-going wave (by selecting any x – t = constant) then, as t increases
indefinitely, we will encounter a breakdown when This leads us to intro-
duce a new variable and otherwise we may use (because we are
following the right-running wave) and we observe that no scaling is associated with
this variable. Thus we transform from (x, t) variables (the near-field) to variables
(the far-field), i.e.

and so our original equation, (3.9), becomes

where

The nature of the appearance of in this equation is identical to the original
equation, suggesting that again we may seek a solution in the form

which gives

and so on. This equation can be integrated once in and, when we invoke decay
conditions at infinity (i.e. all as which mirrors our
given conditions on f(x)), we obtain

This equation is very different from our previous leading-order equation (in the near-
field) : that was simply the classical wave equation. Our dominant equation in the far-
field, (3.21), describes the time evolution of the wave in terms of the wave’s nonlinearity
and its dispersive character Equation (3.21) is a variant of the famous
Korteweg-de Vries (KdV) equation; its solutions, and method of solution, initiated
(from the late 1960s) the important studies in soliton theory. For solutions that decay as
it can be demonstrated that the later terms in the asymptotic expansion
contribute uniformly small corrections to as (This is a far-from-trivial
exercise and is not undertaken in this text; the interested reader who wishes to explore
this further should consult any good text on wave propagation e.g. Whitham, 1974.)
In summary, we have seen that this example (which we have worked through rather
carefully) describes a predominantly linear wave in the near-field (where t = O(1) or
smaller) but, for (the far-field), the wave is, to leading order, described
by a nonlinear equation. Equation (3.21) can be solved exactly and, further, we may