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Finally, the surface boundary condition for gives the equation for identically
cancels—in the form of a variable-coefficient Korteweg-de Vries equation (see E3.1
and Q3.4):

This is usually expressed in terms of to give

which recovers the classical Korteweg-de Vries equation for water waves when we
set D = 1. This equation is the basis for many of the modern studies in water-wave
theory; more background to this, and related problems in water waves, can be found
in Johnson (1997).

5.6 EXTREME THERMAL PROCESSES
This next group of problems concerns phenomena that involve explosions, combus-
tion and the like. The two examples that we will describe are: E5.23 A model for
combustion; E5.24 Thermal runaway.

E5.23 A model for combustion
A model that aims to describe ignition, followed by a rapid combustion, requires a
slow development over a reasonable time scale that precedes a massive change on a
very short time scale, initiated by the attainment of some critical condition. A simple
(non-dimensional) model for such a process (Reiss, 1980) is the equation

where is the concentration of an appropriate chemical that takes part in the
combustive reaction. The whole process is initiated by the small disturbance at
time t = 0. (It should be fairly apparent that this equation can be integrated completely
to give the solution for c, but in implicit form and so the detailed structure as
is far from transparent; this integration is left as an exercise.)
By virtue of the initial value we first seek a solution in the form