Wave propagation in nonlinear media (Q1100342)

The solution of the nonlinear partial differential equation \[ \partial \phi /\partial t+a\phi \quad m\partial \phi /\partial x=b\partial \quad n\phi /\partial x\quad n\quad for\quad t>0,\quad x>0 \] where \(m=1,2\), and \(n=2,3,4\) subject to conditions \(\phi (x,0)=f(x)\) for \(x\geq 0\) and \(\phi (0,t)=g(t)\) and \(\lim_{x\to \infty} \phi (x,t)=0\) for \(t\geq 0\) is carried constants and f(x) and g(t) given sufficiently smooth functions satisfying \(f(0)=g(0)\). (Numerical solutions have appeared previously by \textit{M. N. Oguztöreli}, \textit{E. S. Suhubi}, and \textit{K. V. Leung} [Appl. Math. Comput. 6, 309-334 (1980; Zbl 0424.65061)].) The solution given here is an analytical approximation not involving linearization or perturbation. The appendix summarizes the decomposition method for convenient references. (The book in the 4th reference should have a 1988 date).