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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]

In Part I* we have shown, see Theorem 2.10, that as the coefficient of uxxx tends to zero, the solution of the initial value problem for the KdV equation tends to a limit ū in the distribution sense. We have expressed ū by formula (3.59), where ψx* is the partial derivative with respect to x of the function ψ* defined in Theorem 3.9 as the solution of the variational problem formulated in (2.16), (2.17). ψ* is uniquely characterized by the variational condition (3.34); its partial derivatives satisfy (3.51) and (3.52), where I is the set I0 defined in (3.36). In Section 4 we show that for t < tb, I consists of a single interval, and the ū satisfies ūt − 6$$ \overline {uu} $$x = 0; here tb is the largest time interval in which (12) has a continuous solution. In Section 5 we show that when I consists of a finite number of intervals, ū can be described by Whitham's averaged equation or by the multiphased averaged equations of Flaschka, Forest, and McLaughlin. Equation numbers refer to Part I. [ABSTRACT FROM AUTHOR]