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I n Section 1 we present a general principle for associating nonlinear equations of evolutions with linear operators so that the eigenvalues of the linear operator are integrals of the nonlinear equation. A striking instance of such a procedure is the discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrodinger operator are integrals of the Korteweg-de Vries equation.In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg-de Vries equation, i.e., of solutions which for It( large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of a remarkable series of integrals discovered by Kruskal and Zabusky. 91.In this paper we study the equation We shall study C" solutions of (1.1) defined for all x in (-co, co), which tend to zero as x 4 fa, together with all their x derivatives. It is easy to show that such solutions are uniquely determined by their initial values. Let v be another solution of (1.1) :(1.l)v vt + DUX + vzxx = 0 .Subtracting this from (1.1) and denoting u -v by w , we obtain the linear equation Wt + uwx + wvx + wzxx = 0

In this paper a wide class of difference equations is described for approximating discontinuous time dependent solutions, with prescribed initial data, of hyperbolic systems of nonlinear conservation laws. Among these schemes we determine the best ones, i.e., those which have the smallest truncation error and in which the discontinuities are confined to a narrow bajid of 2-3 meshpoints. These schemes are tested for stability and are found to be stable under a mild strengthening of the CourantFriedrichs-Lewy criterion. Test calculations of one-dimensional flows of compressible fluids with shocks, rarefaction waves and contact discontinuities show excellent agreement with exact solutions. In particular, when Lagrange coordinates are used, there is no smearing of interfaces.The additional terms introduced into the difference scheme for the purpose of keeping the shock transition narrow are similar to, although not identical with, the artificial viscosity terms, and the like of them introduced by Richtmyer and von Ne\iraann and elaborated by other workers in this field.