Norman J. Zabusky

Korteweg‐deVries Equation and Generalizations. V. Uniqueness and Nonexistence of Polynomial Conservation Laws

The conservation laws derived in an earlier paper for the Korteweg‐deVries equation are proved to be the only ones of polynomial form. An algebraic operator formalism is developed to obtain explicit formulas for them.

A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction

Publisher Summary This chapter discusses the application of a synergetic approach to the problems of nonlinear dispersive wave propagation and interaction. The synergetic approach to nonlinear problems was first formulated by von Neumann and it was enunciated in the works of von Neumann and his collaborators in the mid-forties. The synergetic approach to nonlinear mathematical and physical problems can be defined as the simultaneous use of conventional analysis and computer numerical mathematics to obtain solutions to judiciously posed problems concerning the mathematical and physical content of a set of equations. The chapter discusses the study of wave propagation and interaction in a one-dimensional nonlinear—anharmonic—lattice.

Stroboscopic‐Perturbation Procedure for Treating a Class of Nonlinear Wave Equations

A new perturbation procedure is presented for treating initial‐value problems of nonlinear hyperbolic partial differential equations. The characteristic variables of the partial differential equation and the functions of these variables are expanded in powers of e, and the formal solution is uniformly valid over time intervals O(1/e). The uniform first‐order solution is evaluated for the equation ytt=(1+eyx)yxx, subject to the standing‐wave initial conditions: y(x, 0) = a sin πx, yt(x, 0) = 0. This equation is the lowest continuum limit of an equation for which numerical computations are available. The uniform zero‐order solution breaks down after a time tB = 4/eaπ. A detailed study of the solution is made in the vicinity of the breakdown region of the (x, t) plane, and it demonstrates that the formal solution for yx and yt goes from a single‐valued to a triple‐valued function while yxx and ytt become singular. To compare the solutions with the available numerical computations, the yx and yt waveforms are...

Dynamics of nonlinear lattices I. Localized optical excitations, acoustic radiation, and strong nonlinear behavior

Abstract A one-dimensional lattice of equimass particles coupled to nearest neighbors by nonlinear “linear-plus-quadratic” force laws is excited with initial conditions for which alternate masses are displaced along two smooth curves. This results in an interaction between “acoustic,” low-frequency motions and “optical,” high-frequency motions. A continuum description in terms of a pair of coupled partial differential equations is introduced and analytical solutions obtained are found to agree quantitatively with small-amplitude, short-time, optical-acoustic interactions observed in numerical solutions of the lattice equations. Hence the lattice or discretization phenomenon known as “aliasing” (coupling of optical energies to acoustic energies) can be treated analytically by a continuum description if the energies involved are small. As the strength of the initial amplitudes is increased, this description in terms of two smooth curves becomes invalid, and “three-curve” states appear after a short time. A further increase in the nonlinearity results in a rapid cascade of energy across the entire modal energy spectrum. For the times considered, however, we fail to attain complete equipartition of the spectral energies. Instead, very regular features such as one-, two-, three-, and higher -curve states are observed to occupy a large fraction of the lattice length and are preserved as they propagate along and interact with each other.

Exact Invariants for a Class of Nonlinear Wave Equations

A method is given for obtaining explicitly an infinite number of exact invariants for a physical system described by the coupled set of first‐order hyperbolic partial differential equations ∂ui/∂t=Aij(u0,u1,…un) ∂uj/∂x    (i,j=0,1,…,n). Temporal and spatial invariants are constructed as integrals of temporally and spatially invariant densities T and X, over appropriate spatial and temporal intervals, respectively. For physical systems the energy and momentum densities are temporal invariant densities. These invariant densities are solutions of the hodograph transformed equations corresponding to (A1). For the case n = 1 every invariant density T satisfies an equation in conservation form: (Tu0)t − (Tu1)x = 0. The methods are applied to the equation ytt−(1+eyx) αyxx=0, and a denumerable infinity of invariant densities, each expressible as a polynomial, are calculated in two equivalent cases: the first when (A2) (with α = 1) is expressed in zero diagonal form ut = vx, vt = (1 + eu)ux, where u = yx and v = y...