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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 the scattering transform method is used to study the weak limit of solutions to the initial value problem for the Korteweg-deVries (KdV) equation as the dispersion tends to zero. In that limit the associated Schrödinger operator becomes semiclassical, so the exact scattering data is replaced by its corresponding WKB expressions. Only nonpositive initial data are considered; in that case the limiting reflection coefficient vanishes. The explicit solution of Kay and Moses for the reflectionless inverse transform is then analyzed,. and the weak limit, valid for all time, is characterized by a quadratic minimum problem with constraints. This minimum problem is reduced to a Riemann-Hilbert problem in function theory. In Parts II and III we use function theoretical methods to solve the Riemann-Hilbert problem in terms of solutions to an auxiliary initial value problem. The weak limit satisfies the KdV equation with the dispersive term dropped until its derivatives become infinite. Up to that time the weak limit is a strong L2-limit. At later times the weak limit is locally described by Whitham's averaged equations or, more generally, by the equations found by Flaschka et al. using multiphase averaging. For large times, behavior of the weak limit is studied directly from the minimum problem. [ABSTRACT FROM AUTHOR]