COPYRIGHT 2008, PRINCETON UNIVERSITY PRESS

516 chapter 1987 6 5 4 3 2 121084 t020X40600Figure 19.2 A single two-level waveform at time zero progressively breaks up into eightsolitons (labeled) as time increases. The tallest soliton (1) is narrower and faster in its motion tothe right.We want to understand the unusual water waves that occur in shallow, narrowchannels such as canals [Abar, 93, Tab 89]. The analytic description of this “heapof water” was given by Korteweg and deVries (KdeV) [KdeV 95] with the partialdifferential equation∂u(x, t)∂t+ εu(x, t)∂u(x, t)∂x+ µ ∂3 u(x, t)∂x 3 =0. (19.19)As we discussed in §19.1 in our study of Burgers’ equation, the nonlinear termεu ∂u/∂t leads to a sharpening of the wave and ultimately a shock wave. In contrast,as we discussed in our study of dispersion, the ∂ 3 u/∂x 3 term produces broadening.These together with the ∂u/∂t term produce traveling waves. For the properparameters and initial conditions, the dispersive broadening exactly balances thenonlinear narrowing, and a stable traveling wave is formed.KdeV solved (19.19) analytically and proved that the speed (19.1) given byRussell is in fact correct. Seventy years after its discovery, the KdeV equationwas rediscovered by Zabusky and Kruskal [Z&K 65], who solved it numericallyand found that a cos(x/L) initial condition broke up into eight solitarywaves (Figure 19.2). They also found that the parts of the wave with largeramplitudes moved faster than those with smaller amplitudes, which is why thehigher peaks tend to be on the right in Figure 19.2. As if wonders never cease,−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 516