The Small Dispersion Limit of the Korteweg-deVries Equation. III.

In Parts I and II we have derived explicit formulas for the distribution limit ū of the solution of the KdV equation as the coefficient of uxxx tends to zero. This formula contains n parameters β1..., βn whose values, as well as whose number, depends on x and t. In Section 4 we have shown that for t < tb, n = 1, and the value of β, was determined. In Section 5 we have shown that the parameters βj, satisfy a nonlinear system of partial differential equations. In Part III, Section 6 we show that for t large, n = 3, and we determine the asymptotic behavior of β1, β2, β3, and of ū and $$ \overline {u^2 } $$ , for t large. The explicit formulas show that ū and $$ \overline {u^2 } $$ are O(t−l) and O(t−2) respectively (see formulas (6.2) and (6.24)). In Section 7 we study initial data whose value tends to zero as x → +∞, and to −1 as x → −∞. If one accepts some plausible guesses about the behavior of solutions with such initial data, we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution: $$ \bar u(x,t)\left\{ \begin{gathered} \cong - 1for x < - 6t \hfill \\ = s(x/t)for - 6t < x < 4t \hfill \\ \sim 0for - 4\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{t} < x. \hfill \\ \end{gathered} \right. $$ The function s(ξ) is expressible in terms of complete elliptic integrals; a similar formula is derived for $$ \overline {u^2 } $$ . In Section 8 we indicate how to extend the treatment of this series of papers to multihumped (but still negative) initial data. [ABSTRACT FROM AUTHOR]