The small dispersion limit of the korteweg‐de vries equation. ii

Abstract

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 u in the distribution sense. We have expressed u 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 Io defined in (3.36). In Section 4 we show that for t<tb, I consists of a single interval, and the u satisfies ut — 6uux = 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, u 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.

Differential Geometry and Hydrodynamics of Soliton Lattices

“…We should define the time evolution for the multivalued functions. (The same conclusion may be deduced from the results of Lax, Levermore and Venakides [32,33].) We constructed in the works [18,19] the class of special 1-and 3-valued functions for the investigation of stability of the "dispersive shock wave" which should be realized also as the asymptotics t → ∞.…”

Extremely wideband signal shaping using one- and two-dimensional nonuniform nonlinear transmission lines

“…15 What is intriguing is this: suppose we keep h as an arbitrary, nonzero parameter and solve ͑15͒ analytically, using the inverse scattering method, we obtain a function u h ͑x , t͒. In the work of Lax and Levermore, 16 it was shown that in the zero-dispersion h → 0 limit, the sequence u h ͑x , t͒ does not converge to a solution of Burgers' equation ͑17͒. Therefore, we conclude that the h Ͼ 0 continuum model allows fundamentally different phenomena than the h = 0 model.…”

Long time asymptotics of the Korteweg-de Vries equation

“…This method is a natural generalization of the Kay-Moses determinant [6] and its expansion by Lax-Levermore [7]. We recall that (2.8) Formula (2.7) can be generalized formally in three ways to give more solutions to the KdV equation.…”