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

Abstract

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 ε, and the formal solution is uniformly valid over time intervals O(1/ε). The uniform first-order solution is evaluated for the equationytt=(1+εyx)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/εaπ. 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 decomposed into spatial Fourier modes. The effect of breakdown is manifest in the modal amplitudes ∝ Jn(nT)/nT. The modal amplitudes change their asymptotic behavior, from exponentially decreasing as n → ∞, to algebraically decreasing when t goes from smaller to larger than tB. In the time interval up to breakdown, t < tB, the modal energies are in excellent agreement with the modal energies of the numerical computations, whereas for t > tB they diverge. For t < tB, the total energy calculated from the uniform zero-order solution is conserved and equal to the initial value, EC(0)=2a2π2 ∑ n=1∞[Jn(nT)(nT)]2=12a2π2=EC |t=0,t≤tB. Thus, the lowest-continuum-limit equations describe the dynamics of a discrete model for a finite time. A heuristic discussion is given which suggests that the time of description can be extended beyond tB by including higher spatial derivatives in the continuum model.