Peter D. Lax

Integrals of Nonlinear Equations of Evolution and Solitary Waves

In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure 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 |I| large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.

On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws

This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes. Special attention is given to the Godunov-type schemes that result from using an approximate solution of the Riemann problem. For schemes based on flux splitting, the approximate Riemann solution can be interpreted as a solution of the collisionless Boltzmann equation.

Hyperbolic systems of conservation laws and the mathematical theory of shock waves

Quasi-linear Hyperbolic Equations Conservation Laws Single Conservation Laws The Decay of Solutions as t Tends to Infinity Hyperbolic Systems of Conservation Laws Pairs of Conservation Laws Notes References.

Shock Waves and Entropy

Publisher Summary This chapter provides an overview of shock waves and entropy. It describes systems of the first order partial differential equations in conservation form: ∂ t U + ∂ X F = 0, F = F(u). In many cases, all smooth solutions of the first order partial differential equations in conservation form satisfy an additional conservation law where U is a convex function of u. The chapter discusses that for all weak solutions of ∂ t u j +∂ x f j = 0, j=1,…, m, f j =f j (u 1 ,…, u m ), which are limits of solutions of modifications ∂ t u j +∂ x f j = 0, j=1,…, m, f j =f j (u 1 ,…, u m ) , by the introduction of various kinds of dissipation, satisfy the entropy inequality, that is, ∂ t U + ∂ x F≦ 0. The chapter also explains that for weak solutions, which contain discontinuities of moderate strength, ∂ t U + ∂ x F≦ 0 is equivalent to the usual shock condition involving the number of characteristics impinging on the shock. The chapter also describes all possible entropy conditions of ∂ t U + ∂ x F≦ 0 that can be associated to a given hyperbolic system of two conservation laws.

Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations

In a recent paper Zabusky has given an accurate estimate of the time interval in which solutions of the nonlinear string equation ytt = c2(1 + eyx)yxx exist. A previous numerical study of solutions of this equation disclosed an anomaly in the partition of energy among the various modes; Zabuskys estimate shows that at the time when the anomaly was observed the solution does not exist. The proof of Zabusky uses the hodograph method; in this note we give a much simpler derivation of the same result based on an estimate given some years ago by the author.

On Finite-Difference Approximations and Entropy Conditions for Shocks.

Weak solutions of hyperbolic conservation laws are not uniquely determined by their initial values; an entropy condition is needed to pick out the physically relevant solution. The question arises whether finite-difference approximations converge to this particular solution. It is shown in this paper that, in the case of a single conservation law, monotone schemes, when convergent, always converge to the physically relevant solution. Numerical examples show that this is not always the case with nonmonotone schemes, such as the Lax--Wendroff scheme. 4 figures, 2 tables. (auth)

The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces

The counting numbers for discrete subgroups of motions in Euclidean and non-Euclidean spaces are obtained using the wave equation as the principal tool. In dimensions 2 and 3 the error estimates are close to the best known.

Proof of a conjecture of P. Erdös on the derivative of a polynomial

Introduction. We start out from the following consequence of S. Bernsteins well known theorem on trigonometric polynomials. Let pn(z) be a polynomial of degree n for which | ^(^) | ^ 1 holds as \z\ gal; then I/>n (s)| ^n as \z\ g l with |^n(«)| —n if and only if pn(z)=az , \a\ = 1. Some time ago P. Erdös conjectured that if \pn(z)\ ^ 1 holds as |z\ ^ 1 and pn(z) has no roots inside the unit circle, then \pl (z)\ ^n/2 as \z\ g l . In the present note we give a proof of this conjecture.

Almost Periodic Solutions of the KdV Equation

In this talk we discuss the almost periodic behavior in time of space periodic solutions of the KdV equation \[ u_t + uu_x + u_{xxx} = 0.\] We present a new proof, based on a recursion relation of Lenart, for the existence of an infinite sequence of conserved functionals

A Random Choice Finite Difference Scheme for Hyperbolic Conservation Laws

F_n (u)